robotic control lecture 1 dynamics and modeling. robotic control2 a brief history… started as a...

Robotic Control

Lecture 1

Dynamics and Modeling

Robotic Control 2

A brief history…

Started as a work of fiction

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

Robotic Control 3

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

Robotic Control 4

Robotics

Modern robots

MechanicsDigital Computation

Electronic Sensors

Path Planning Learning/Adaptation

Coordination

Actuation

Robotic Control 5

Types of Robots

Industrial Locomotion/Exploration Medical Home/Entertainment

Robotic Control 6

Industrial Robots

Assembly of an automobile

Drilling/ Welding/Cutting

Coating/Painting

Robotic Control 7

Locomotion/Exploration

Space Exploration

Underwater exploration

Robo-Cop

Robotic Control 8

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

Robotic Control 9

House-hold/Entertainment

AsimoToys

Robotic Control 10

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

Robotic Control 11

Mathematical Modeling

Local vs. Global coordinatesTranslate from joint angles to end position

Jacobiancoordinate transforms linearization

Kinematics Dynamics

Robotic Control 12

Mechanics of Multi-link arms

Local vs. Global coordinates Coordinate Transforms

Jacobians Kinematics

Robotic Control 13

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

Robotic Control 14

Coordinate Transformation Cntd.

Homogeneous transformationMatrix of partial

derivativesTransforms joint

angles (q) into manipulator coordinates Jqx

n

1

q

Robotic Control 15

Coordinate Transformation

2-link arm, relative coordinates

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

Robotic Control 16

Coordinate Transformation Step 2: Take

partial derivatives to find J

Robotic Control 17

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

Robotic Control 18

Finding the Dynamic Model of a Robotic System

Dynamics Lagrange Method

Equations of Motion MATLAB Simulation

Robotic Control 19

Step 1: Identify Model Mechanics

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

Example: 2-link robotic arm

Robotic Control 20

Step 2: Identify Parameters

For each link, find or calculateMass, mi

Length, liCenter of gravity, lCi

Moment of Inertia, ii

m1

i1=m1l12 / 3

Robotic Control 21

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

Robotic Control 22

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

Robotic Control 23

Energy Sums (2-Link Arm)

T = sum of kinetic energies:

V = sum of potential energies:

Robotic Control 24

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)

Robotic Control 25

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

Robotic Control 26

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

Robotic Control 27

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)

Robotic Control 28

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);

Robotic Control 29

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')

Robotic Control 30

Open Loop Model ValidationZero State/Input

Arm falls down and settles in that position

Robotic Control 31

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

Robotic Control 32

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

Robotic Control 33

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

Robotic Control 34

Robotic Control

Path Generation Displacement Control

Force Control Hybrid Control

Robotic Control 35

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

Robotic Control 36

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

Robotic Control 37

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

Robotic Control 38

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

Robotic Control 39

Next Time…

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