Robotic Control

Lecture 1

Dynamics and Modeling

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A brief history…

Started as a work of fiction

Czech playwright Karel Capek coined the term robot in his play Rossum’s Universal Robots

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Numerical control

Developed after WWII and were designed to perform specific tasks

Instruction were given to machines in the form of numeric codes (NC systems)

Typically open-loop systems, relied on skill of programmers to avoid crashes

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Robotics

Modern robots

MechanicsDigital Computation

Electronic Sensors

Path Planning Learning/Adaptation

Coordination

Actuation

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Types of Robots

Industrial Locomotion/Exploration Medical Home/Entertainment

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Industrial Robots

Assembly of an automobile

Drilling/ Welding/Cutting

Coating/Painting

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Locomotion/Exploration

Space Exploration

Underwater exploration

Robo-Cop

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Medical

a) World's first CE-marked medical robot for head surgeryb) Surgical robot used in spine surgery, redundant manual guidance. c) Autoclavable instrument guidance (4 DoF) for milling, drilling, endoscope

guidance and biopsy applications

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House-hold/Entertainment

AsimoToys

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Purpose of Robotic Control

Direct control of forces or displacements of a manipulator

Path planning and navigation (mobile robots)

Compensate for robot’s dynamic properties (inertia, damping, etc.)

Avoid internal/external obstacles

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Mathematical Modeling

Local vs. Global coordinatesTranslate from joint angles to end position

Jacobiancoordinate transforms linearization

Kinematics Dynamics

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Mechanics of Multi-link arms

Local vs. Global coordinates Coordinate Transforms

Jacobians Kinematics

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Local vs. Global Coordinates

Local coordinates Describe joint angles or extensionSimple and intuitive description for each link

Global CoordinatesTypically describe the end effector /

manipulator’s position and angle in space“output” coordinates required for control of

force or displacement

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Coordinate Transformation Cntd.

Homogeneous transformationMatrix of partial

derivativesTransforms joint

angles (q) into manipulator coordinates Jqx

n

1

q

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Coordinate Transformation

2-link arm, relative coordinates

Step 1: Define x and y in terms of θ1 and θ2

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Coordinate Transformation Step 2: Take

partial derivatives to find J

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Joint Singularities Singularity condition Loss of 1 or more DOF J becomes singular 21

xx

Occurs at: Boundaries of

workspace Critical points (for

multi-link arms

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Finding the Dynamic Model of a Robotic System

Dynamics Lagrange Method

Equations of Motion MATLAB Simulation

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Step 1: Identify Model Mechanics

Source: Peter R. Kraus, 2-link arm dynamics

Example: 2-link robotic arm

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Step 2: Identify Parameters

For each link, find or calculateMass, mi

Length, liCenter of gravity, lCi

Moment of Inertia, ii

m1

i1=m1l12 / 3

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Step 3: Formulate Lagrangian

Lagrangian L defined as differencebetween kinetic and potential energy:

L is a scalar function of q and dq/dt L requires only first derivatives in time

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Kinetic and Potential Energies

iiCiii hglmV 0)sin(

Kinetic energy of individual links in an n-link arm

Potential energy of individual links

Height of link end

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Energy Sums (2-Link Arm)

T = sum of kinetic energies:

V = sum of potential energies:

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Step 4: Equations of Motion

Calculate partial derivatives of L wrt qi, dqi/dt and plug into general equation:

Inertia

(d2qi/dt2)

Conservative Forces

Non-conservative Forces

(damping, inputs)

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M – Inertia Matrix Positive Definite Configuration dependent Non-linear terms: sin(θ), cos(θ)

C – Coriolis forces Non-linear terms: sin(θ), cos(θ),

(dθ/dt)2, (dθ/dt)*θ

Fg – Gravitational forces Non-linear terms: sin(θ), cos(θ)

Equations of Motion – Structure

Source: Peter R. Kraus, 2-link arm dynamics

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Equations of Motion for 2-Link Arm, Relative coordinates

Source: Peter R. Kraus, 2-link arm dynamics

Coriolis forces, c(θi,dθi/dt) Conservative forces

(gravity)

M- Inertia matrix

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Alternate Form: Absolute Joint Angles If relative coordinates are

written as θ1’,θ2’, substitute θ1=θ1’ and θ2=θ2’+θ1’

Advantages: M matrix is now symmetric Cross-coupling of eliminated from C, from F matrices Simpler equations (easier to check/solve)

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Matlab Codefunction xdot= robot_2link_abs(t,x)global T%parametersg = 9.8;m = [10, 10];l = [2, 1];%segment lengths l1, l2lc =[1, 0.5]; %distance from centeri = [m(1)*l(1)^2/3, m(2)*l(2)^2/3]; %moments of inertia i1, i2, need to validate coef'sc=[100,100];xdot = zeros(4,1); %matix equations M= [m(2)*lc(1)^2+m(2)*l(1)^2+i(1), m(2)*l(1)*lc(2)^2*cos(x(1)-x(2)); m(2)*l(1)*lc(2)*cos(x(1)-x(2)),+m(2)*lc(2)^2+i(2)];

C= [-m(2)*l(1)*lc(2)*sin(x(1)-x(2))*x(4)^2; -m(2)*l(1)*lc(2)*sin(x(1)-x(2))*x(3)^2];

Fg= [(m(1)*lc(1)+m(2)*l(1))*g*cos(x(1)); m(2)*g*lc(2)*cos(x(2))];

T =[0;0]; % input torque vector

tau =T+[-x(3)*c(1);-x(4)*c(2)]; %input torques, xdot(1:2,1)=x(3:4);xdot(3:4,1)= M\(tau-Fg-C);

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Matlab Code

t0=0;tf=20;

x0=[pi/2 0 0 0];

[t,x] = ode45('robot_2link_abs',[t0 tf],x0);

figure(1)

plot(t,x(:,1:2))

Title ('Robotic Arm Simulation for x0=[pi/2 0 0 0]and T=[sin(t);0] ')

legend('\theta_1','\theta_2')

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Open Loop Model ValidationZero State/Input

Arm falls down and settles in that position

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Open Loop - Static Equilibrium

x0= [-pi/2 –pi/2 0 0]x0= [-pi/2 pi/2 0 0]

x0= [pi/2 pi/2 0 0]x0= [pi/2 -pi/2 0 0]

Arm does not change its position- Behavior is as expected

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Open Loop - Step ResponseTorque applied to first joint

Torque applied to second joint

When torque is applied to the first joint, second joint falls down

When torque is applied to the second joint, first joint falls down

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Input (torque) as Sine functionTorque applied to first joint

Torque applied to first joint

When torque is applied to the first joint, the first joint oscillates and the second follows it with a delay

When torque is applied to the second joint, the second joint oscillates and the first follows it with a delay

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Robotic Control

Path Generation Displacement Control

Force Control Hybrid Control

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To find desired joint space trajectory qd(t) given the desired Cartesian trajectory using inverse kinematics

Given workspace or Cartesian trajectory

in the (x, y) plane which is a function of time t. Arm control, angles θ1, θ2, Convenient to convert the specified Cartesian

trajectory (x(t), y(t)) into a joint space trajectory (θ1(t), θ2(t))

Path Generation

( ) ( ), ( )p t x t y t

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Trajectory Control Types

Displacement ControlControl the displacement i.e. angles or

positioning in spaceRobot Manipulators

Adequate performance rigid body Only require desired trajectory movement Examples:

Moving Payloads Painting Objects

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Trajectory Control Types (cont.)

Force Control – Robotic ManipulatorRigid “stiff” body makes if difficultControl the force being applied by the

manipulator – set-point controlExamples:

Grinding Sanding

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Trajectory Control Types (cont.)

Hybrid Control – Robot Manipulator Control the force and position of the manipulator Force Control, set-point control where end effector/

manipulator position and desired force is constant. Idea is to decouple the position and force control

problems into subtasks via a task space formulation. Example:

Writing on a chalk board

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Next Time…

Path Generation Displacement (Position) Control Force Control Hybrid Control i.e. Force/Position Feedback Linearization Adaptive Control Neural Network Control 2DOF Example