Video 1.1 Vijay Kumar and Ani Hsieh 1
Robotics: Dynamics and Control Vijay Kumar and Ani Hsieh University of Pennsylvania 2
Why? Robots live in a physical world The physical world is governed by the laws of motion Fundamental understanding of dynamics of robots 3
The Goal Models of robots Robot manipulators, ground robots, flying robots Beyond geometric and kinematic models to dynamic models Use dynamic models for real world applications 4
Dynamics Two sets of problems: Forward dynamics How do robots move when we apply forces or torques to the actuators, or currents/voltages to the motors? Inverse dynamics What forces or torques or currents or voltages to apply to achieve a desired output (force or moment or velocity or acceleration)? 5
Robotic Manipulation 6
Cooperative Manipulation 7
Coordination of Ground Robots 8
Autonomous Quadrotors 9
Video 1.2 Vijay Kumar and Ani Hsieh 10
Dynamics and Control Introduction Vijay Kumar and Ani Hsieh University of Pennsylvania 11
What should you know? 3-D vectors, geometry Vector calculus, kinematics Rotation matrices Transformation matrices 12
What you will learn How to create dynamic models of robots? How to simulate robotic systems? How to control robotic systems? 13
Dynamics Particle dynamics Kinematics Kinetics Rigid body dynamics Kinematics Kinetics Application to chains of rigid bodies Newton-Euler Equations of Motion Lagrange s Equations of Motion 14
Simulation Forward Dynamics How does the robot move if you apply a set of forces or torques at the actuators http://money.cnn.com/2015/04/07/technology/sa wyer-robot-manufacturing-revolution/ J. Thomas, G. Loianno, J. Polin, K. Sreenath, and V. Kumar, Toward autonomous avianinspired grasping for micro aerial vehicles, Bioinspiration and Biomimetics, vol. 9, no. 2, p. 025010, June 2014. 15
Control Inverse Dynamics What forces or torques need to be applied by the actuators in order to get the robot to move or act in a desired manner http://money.cnn.com/2015/04/07/technology/sa wyer-robot-manufacturing-revolution/ J. Thomas, G. Loianno, J. Polin, K. Sreenath, and V. Kumar, Toward autonomous avianinspired grasping for micro aerial vehicles, Bioinspiration and Biomimetics, vol. 9, no. 2, p. 025010, June 2014. 16
Applications Robot manipulators http://ros.org Ground robots: wheeled Flying robots: quadrotors 17
Video 1.3 Vijay Kumar and Ani Hsieh 18
Dynamics and Control Review Vijay Kumar and Ani Hsieh University of Pennsylvania 19
b2 Reference Frames b1 Reference frame Origin b3 P B Basis vectors Reference frame Origin a1 a2 Basis vectors O A a3 20
b2 Position Vectors b1 Reference frame Origin Q b3 P B Basis vectors Position Vectors Position vectors for P and Q in A a1 a2 Position vector of Q in B O A a3 21
Position Vectors Position vectors for P and Q in A Q P B a1 a2 O A a3 22
b2 Transformations b1 Reference frames Origins Q b3 P B Basis vectors Rigid Body Transformation Position vector of Q in A a1 a2 O Position vector Q in B A a3 23
b2 Transformations b1 Q b3 P B a1 a2 O A a3 24
b2 Transformations b1 Q b3 P B a1 a2 O A Rotation Matrix a3 25
b2 Rotation Matrix b1 b3 B a1 a2 a3 26
b Homogeneous Transformation Matrix 2 b1 Q b3 P B a1 a2 Position of Q in A Position of Q in B 4x4 homogeneous transformation matrix O A a3 27
Video 1.4 Vijay Kumar and Ani Hsieh 28
Dynamics and Control Velocity and Acceleration Analysis Vijay Kumar and Ani Hsieh University of Pennsylvania 29
b2 Position Vectors b1 Reference frame Origin Q b3 P B Basis vectors Position Vectors Position vectors for P and Q in A a1 a2 Position vector of Q in B O A a3 30
Velocity Vectors Velocity of P and Q in A Q P B a1 a2 O A a3 31
Velocity Vectors Velocity of P and Q in B Q P B Zero, since both points are fixed to B! a1 a2 O A a3 32
Velocity Vectors Velocity of P and Q in A Q P B a1 a2 O How to relate velocities of two points fixed to the same rigid body? A a3 33
b2 Recall b1 Q b3 P B a1 a2 O A a3 34
Velocities of 2 points fixed to the same rigid body b2 b1 Q b3 P B a1 a2 O A a3 3x3 skew symmetric 35
Example: Rotation about a single axis Rotation about the z-axis through q ya xb yb q xa za 36
Velocities of 2 points fixed to the same rigid body b2 b1 Q b3 P B Recall a 3x3 skew symmetric matrix encodes a cross product operation a1 a2 O A a3 37
Acceleration Analysis Acceleration of P and Q in A Q P B a1 a2 O A a3 38
Two Approaches Lagrangian Mechanics need expressions of kinetic and potential energy, and external forces/moments Q P B Newtonian Mechanics need expressions for accelerations and external forces/moments a1 a2 O A a3 39
Video 1.5 Vijay Kumar and Ani Hsieh 40
Dynamics and Control Velocity and Acceleration Analysis: Examples Vijay Kumar and Ani Hsieh University of Pennsylvania 41
A one link manipulator Inertial reference frame E Origin O Basis vectors P is fixed to link 1 x0 O θ1 P a1 y0 Joint 42
Position and Velocity Vectors x0 O θ1 P a1 y0 Joint 43
Two Link Manipulator Inertial reference frame E Origin O Basis vectors P is fixed to both links 1 and 2 P and Q are fixed to link 2 θ x0 1 Joint 2 x1 a1 θ2 Link 2 P Link 1 O y0 Q y1 a2 x2 y2 Joint 1 44
Position Vectors x1 θ x0 1 a1 θ2 Link 2 P Link 1 O y0 Q y1 a2 x2 y2 45
Position Vectors x1 θ x0 1 a1 θ2 Link 2 P Link 1 O y0 Q y1 a2 x2 y2 46
Velocity of point Q in the inertial frame x1 θ x0 1 a1 θ2 Link 2 P Link 1 O y0 Q y1 a2 x2 y2 47
Velocity of point Q in the inertial frame alternative approach x1 θ2 P x0 θ 1 Q y1 x2 O y0 y2 48