Robotic Control Lecture 1 Dynamics and Modeling Robotic

Robotic Control Lecture 1 Dynamics and Modeling Robotic Control

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 2

Numerical control n Developed after WWII and were designed to perform specific tasks n 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 n Robotic Control 3

Modern robots Digital Computation Mechanics Coordination Electronic Sensors Actuation Path Planning Learning/Adaptation Robotics Robotic Control 4

Types of Robots Industrial n Locomotion/Exploration n Medical n Home/Entertainment n Robotic Control 5

Industrial Robots Coating/Painting Assembly of an automobile Robotic Control Drilling/ Welding/Cutting 6

Locomotion/Exploration Underwater exploration Space Exploration Robo-Cop Robotic Control 7

Medical 1. a) World's first CE-marked medical robot for head surgery 2. b) Surgical robot used in spine surgery, redundant manual guidance. c) Autoclavable instrument guidance (4 Do. F) for milling, drilling, endoscope guidance and biopsy applications Robotic Control 8

House-hold/Entertainment Toys Asimo Robotic Control 9

Purpose of Robotic Control Direct control of forces or displacements of a manipulator n Path planning and navigation (mobile robots) n Compensate for robot’s dynamic properties (inertia, damping, etc. ) n Avoid internal/external obstacles n Robotic Control 10

Mathematical Modeling n Local vs. Global coordinates ¨ Translate n from joint angles to end position Jacobian ¨ coordinate transforms ¨ linearization Kinematics n Dynamics n Robotic Control 11

Mechanics of Multi-link arms n Local vs. Global coordinates n Coordinate Transforms n Jacobians n Kinematics Robotic Control 12

Local vs. Global Coordinates n Local coordinates ¨ Describe joint angles or extension ¨ Simple and intuitive description for each link n Global Coordinates ¨ Typically describe the end effector / manipulator’s position and angle in space ¨ “output” coordinates required for control of force or displacement Robotic Control 13

Coordinate Transformation Cntd. n Homogeneous transformation ¨ Matrix of partial derivatives ¨ Transforms joint angles (q) into manipulator coordinates Robotic Control 14

Coordinate Transformation 2 -link arm, relative coordinates n Step 1: Define x and y in terms of θ 1 and θ 2 n Robotic Control 15

Coordinate Transformation n Step 2: Take partial derivatives to find J Robotic Control 16

Joint Singularities n n n Singularity condition Loss of 1 or more DOF J becomes singular n Occurs at: ¨ Boundaries of workspace ¨ Critical points (for multi-link arms Robotic Control 17

Finding the Dynamic Model of a Robotic System Dynamics n Lagrange Method n Equations of Motion n MATLAB Simulation n Robotic Control 18

Step 1: Identify Model Mechanics Example: 2 -link robotic arm Source: Peter R. Kraus, 2 -link arm dynamics Robotic Control 19

Step 2: Identify Parameters n For each link, find or calculate ¨ Mass, mi ¨ Length, li ¨ Center of gravity, l. Ci ¨ Moment of Inertia, ii m 1 i 1=m 1 l 12 / 3 Robotic Control 20

Step 3: Formulate Lagrangian n Lagrangian L defined as difference between kinetic and potential energy: L is a scalar function of q and dq/dt n L requires only first derivatives in time n Robotic Control 21

Kinetic and Potential Energies n Kinetic energy of individual links in an n-link arm n Potential energy of individual links Height of link end Robotic Control 22

Energy Sums (2 -Link Arm) n T = sum of kinetic energies: n V = sum of potential energies: Robotic Control 23

Step 4: Equations of Motion n Calculate partial derivatives of L wrt qi, dqi/dt and plug into general equation: Inertia (d 2 qi/dt 2) Conservative Forces Non-conservative Forces (damping, inputs) Robotic Control 24

Equations of Motion – Structure n M – Inertia Matrix ¨ Positive Definite ¨ Configuration dependent ¨ Non-linear terms: sin(θ), cos(θ) n C – Coriolis forces ¨ Non-linear terms: sin(θ), cos(θ), (dθ/dt)2, (dθ/dt)*θ n Fg – Gravitational forces ¨ Non-linear terms: sin(θ), cos(θ) Source: Peter R. Kraus, 2 -link arm dynamics Robotic Control 25

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) Robotic Control 26

Alternate Form: Absolute Joint Angles n n If relative coordinates are written as θ 1’, θ 2’, substitute θ 1=θ 1’ and θ 2=θ 2’+θ 1’ Advantages: ¨ ¨ ¨ M matrix is now symmetric Cross-coupling of eliminated from C, from F matrices Simpler equations (easier to check/solve) Robotic Control 27

Matlab Code function xdot= robot_2 link_abs(t, x) global T %parameters g = 9. 8; m = [10, 10]; l = [2, 1]; %segment lengths l 1, l 2 lc =[1, 0. 5]; %distance from center i = [m(1)*l(1)^2/3, m(2)*l(2)^2/3]; %moments of inertia i 1, i 2, 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); Robotic Control 28

Matlab Code t 0=0; tf=20; x 0=[pi/2 0 0 0]; [t, x] = ode 45('robot_2 link_abs', [t 0 tf], x 0); figure(1) plot(t, x(: , 1: 2)) Title ('Robotic Arm Simulation for x 0=[pi/2 0 0 0]and T=[sin(t); 0] ') legend('theta_1', 'theta_2') Robotic Control 29

Open Loop Model Validation Zero State/Input q Arm falls down and settles in that position Robotic Control 30

Open Loop - Static Equilibrium x 0= [-pi/2 –pi/2 0 0] x 0= [pi/2 0 0] x 0= [-pi/2 0 0] x 0= [pi/2 -pi/2 0 0] q. Arm does not change its position- Behavior is as expected Robotic Control 31

Open Loop - Step Response Torque applied to second joint Torque applied to first joint q When torque is applied to the first joint, second joint falls down q When torque is applied to the second joint, first joint falls down Robotic Control 32

Input (torque) as Sine function Torque applied to first joint q When torque is applied to the first joint, the first joint oscillates and the second follows it with a delay q When torque is applied to the second joint, the second joint oscillates and the first follows it with a delay Robotic Control 33

Robotic Control Path Generation Displacement Control n Force Control n Hybrid Control n n Robotic Control 34

Path Generation n n 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 θ 1, θ 2, Convenient to convert the specified Cartesian trajectory (x(t), y(t)) into a joint space trajectory (θ 1(t), θ 2(t)) Robotic Control 35

Trajectory Control Types n Displacement Control ¨ Control the displacement i. e. angles or positioning in space ¨ Robot Manipulators Adequate performance rigid body n Only require desired trajectory movement n Examples: n Moving Payloads ¨ Painting Objects ¨ Robotic Control 36

Trajectory Control Types (cont. ) n Force Control – Robotic Manipulator ¨ Rigid “stiff” body makes if difficult ¨ Control the force being applied by the manipulator – set-point control ¨ Examples: Grinding n Sanding n Robotic Control 37

Trajectory Control Types (cont. ) n 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: n Writing on a chalk board Robotic Control 38

Next Time… n n n n Path Generation Displacement (Position) Control Force Control Hybrid Control i. e. Force/Position Feedback Linearization Adaptive Control Neural Network Control 2 DOF Example Robotic Control 39