1. Lecture 1 Dynamics and Modeling
Robotic Control
Lecture 1
Dynamics and Modeling

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
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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
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4. Modern robots Mechanics Digital Computation Coordination
Electronic Sensors
Actuation
Path Planning
Learning/Adaptation
Robotics
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5. Types of Robots Industrial Locomotion/Exploration Medical
Home/Entertainment
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6. Industrial Robots Coating/Painting Assembly of an automobile
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Drilling/ Welding/Cutting

7. Locomotion/Exploration
Underwater exploration
Space Exploration
Robo-Cop
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8. Medical a) World's first CE-marked medical robot for head surgery
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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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9. House-hold/Entertainment
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Toys
Asimo
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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
Updated, added more detail
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11. Mathematical Modeling
Local vs. Global coordinates
Translate from joint angles to end position
Jacobian
coordinate transforms
linearization
Kinematics
Dynamics
Rearranged to make more sense
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12. Mechanics of Multi-link arms
Local vs. Global coordinates
Coordinate Transforms
Jacobians
Kinematics
Added outline
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13. 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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14. Coordinate Transformation Cntd.
Homogeneous transformation
Matrix of partial derivatives
Transforms joint angles (q) into manipulator coordinates
Added additional explanationa nd equations
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15. Coordinate Transformation
2-link arm, relative coordinates
Step 1: Define x and y in terms of θ1 and θ2
Broke down coordinate transformation into steps
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16. Coordinate Transformation
Step 2: Take partial derivatives to find J
Showed outcome of Jacobian calculation
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17. Joint Singularities Occurs at: Singularity condition
Loss of 1 or more DOF
J becomes singular
Occurs at:
Boundaries of workspace
Critical points (for multi-link arms
Explain importance of singularity in terms of control issues
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18. Finding the Dynamic Model of a Robotic System
Dynamics
Lagrange Method
Equations of Motion
MATLAB Simulation
Added outline
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19. Step 1: Identify Model Mechanics
Example: 2-link robotic arm
Add text descriptions of parts
Source: Peter R. Kraus, 2-link arm dynamics
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20. Step 2: Identify Parameters
For each link, find or calculate
Mass, mi
Length, li
Center of gravity, lCi
Moment of Inertia, ii m1
Explain effects of varying link size and cross section
i1=m1l12 / 3
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21. 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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22. Kinetic and Potential Energies
Kinetic energy of individual links in an n-link arm
Potential energy of individual links
Explain parts of equation
Height of
link end
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23. Energy Sums (2-Link Arm)
T = sum of kinetic energies:
V = sum of potential energies:
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24. Step 4: Equations of Motion
Calculate partial derivatives of L wrt qi, dqi/dt and plug into general equation:
Non-conservative Forces
(damping, inputs)
Inertia
(d2qi/dt2)
Conservative Forces
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25. 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
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Source: Peter R. Kraus, 2-link arm dynamics

26. Equations of Motion for 2-Link Arm, Relative coordinates
M- Inertia matrix
Conservative forces
(gravity)
Coriolis forces, c(θi,dθi/dt)
Point out non-lienarities
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Source: Peter R. Kraus, 2-link arm dynamics

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)
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28. Matlab Code Robotic Control 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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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/ ]and T=[sin(t);0] ')
legend('\theta_1','\theta_2')
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30. Open Loop Model Validation
Zero State/Input
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Arm falls down and settles in that position
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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
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32. 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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33. 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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34. Robotic Control Path Generation Displacement Control Force Control
Hybrid Control
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35. Path Generation
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))
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36. 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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37. 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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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
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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
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