Robotic Control
Lecture 1
Dynamics and Modeling
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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Modern robots
Mechanics
Digital Computation
Coordination
Electronic Sensors
Actuation
Path Planning
Learning/Adaptation
Robotics
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Types of Robots
Industrial
 Locomotion/Exploration
 Medical
 Home/Entertainment

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Industrial Robots
Coating/Painting
Assembly of an automobile
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Drilling/ Welding/Cutting
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Locomotion/Exploration
Underwater exploration
Space Exploration
Robo-Cop
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Medical
a) World's first CE-marked medical robot for head surgery
b) 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
Toys
Asimo
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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 coordinates
 Translate

from joint angles to end position
Jacobian
 coordinate
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 extension
 Simple and intuitive description for each link

Global Coordinates
 Typically
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
transformation
 Matrix
of partial
derivatives
 Transforms joint
angles (q) into
1 
manipulator

q

coordinates
 
 n 
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x  Jq
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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
x
x

1  2

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
Example: 2-link robotic arm
Source: Peter R. Kraus, 2-link arm dynamics
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Step 2: Identify Parameters

For each link, find or calculate
 Mass,
mi
 Length, li
 Center of gravity, lCi
 Moment of Inertia, ii
m1
i1=m1l12 / 3
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Step 3: Formulate Lagrangian

Lagrangian L defined as difference
between 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

Kinetic energy of individual links in an n-link arm

Potential energy of individual links
Vi  mi lCi g sin( i )h0i
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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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Equations of Motion – Structure

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)
Source: Peter R. Kraus, 2-link arm dynamics
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Equations of Motion for 2-Link Arm,
Relative coordinates
M- Inertia matrix
Coriolis forces, c(θi,dθi/dt)
Source: Peter R. Kraus, 2-link arm dynamics
Conservative forces
(gravity)
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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 Code
function xdot= robot_2link_abs(t,x)
global T
%parameters
g = 9.8;
m = [10, 10];
l = [2, 1];%segment lengths l1, l2
lc =[1, 0.5]; %distance from center
i = [m(1)*l(1)^2/3, m(2)*l(2)^2/3]; %moments of inertia i1, i2, need to validate coef's
c=[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 Validation
Zero 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 Response
Torque applied to second joint
Torque applied to first 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 function
Torque 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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Path Generation




To find desired joint space trajectory qd(t) given
the desired Cartesian trajectory using inverse
kinematics
Given workspace or Cartesian trajectory
p (t )   x(t ), y (t ) 
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))
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Trajectory Control Types

Displacement Control
 Control
the displacement i.e. angles or
positioning in space
 Robot 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 Manipulator
 Rigid
“stiff” body makes if difficult
 Control the force being applied by the
manipulator – set-point control
 Examples:
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…
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






Path Generation
Displacement (Position) Control
Force Control
Hybrid Control i.e. Force/Position
Feedback Linearization
Adaptive Control
Neural Network Control
2DOF Example
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