산선생의 집입니다. 환영해요

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1 Biped Walking Robot Biped Walking Robot Simulation Program Down(Visual Studio 6.0 ) ). Version.,. Biped Walking Robot - Project Degree of Freedom : 12(,,, 12) :,, : Link.

2 Kinematics. 1. Z (~ Diablo Set Item ) 2. X Z(I-1) Z(I) (Craig Index ) 3. Y 4. Base [0] [1] Parameter

3 New Kinematics, Dynamics, ZMP Dynamics.,,. New. [F], Base [B1] [B2],. Kinematics [B1] Top-down, [B2] Bottom-Up. Inverse Kinematics Analytic. Link Parameters Table Links Denavit-Hartenberg notation Links 4 Parameter Project Table. Parameter Craig. ) Index, Degree Link α(i-1) a (i-1) d (i) Θ(i) [1] θ1(0) [2] θ2(-90) [3] θ3(0) [4] 0 L2 0 θ4(0) [5] 0 L2 0 θ5(90) [6] θ6(0)

4 [ 1] Top Down Link Parameters Table Forward Kinematics Forward Kinematics Theta(Joint angle) End Effector (Cartesian Space) Table Forward Kinematics. [0] [1] Transformation Matrix [1] [2] Trasformation Matrix.. [5] [6] Post-Multiplication [0] [6] Trasformation Matrix. [i-1] [I] Transformation Matrix. Inverse Kinematics Inverse Kinematics End Effector (Cartesian Space), [0] [6] Trasformation Matrix Theta(Joint angle)

5 New Inverse,.. Inverse Kinematics Analytical Solutions Numerical Solutions. Analytical Algorithm -. ==> Complete Numerical. - Numerical reliable ==> Error. - Closed-Form Algebraic Elimination. - Closed-Form Closed-Form. - Closed-Form 6 System Closed-Form. - Algebraic Elimination Multivariable polynomial. Numerical Algorithm -. - Analytical. - Analytical reliable ==>. - System ill-posed. Analytical ill-posed Newton-Raphson Algorithm for solving systems of nonlinear equations 2. Inverse Kinematics Differential Equations

6 3. Inverse Kinematics nonlinear optimization Parallel Mechanism VS Serial Mechanism Parallel Mechanism : Inverse Kinematics Solution. Forward Kinematics Solutions. Serial Mechanism : Inverse Kinematics Solutions, Forward Kinematics Solution. Kinematics End-Effector. Trajectory Plan Robot. Jacobian,, Trajectory Trajectory Trajectory Plan 1. Actuators 2. Biped Walking Robot 3. Path 4. System. 5. System Trajectory Scheme Joint Space Cartesian Space. Joint Space Trajectory Scheme Trajectory Polynomial 4-3-4, Five Cubic Smooth

7 5 Polynomial. : angle,, angle,, ==> 6 5 Polynomial Sampling Time Polynomial Joint Angles. Control Input. Path End-Effector. Inverse Kinematics Trajectory. Joint Level Trajectory Robot,., Trajectory Cartesian Level. Cartesian Space Trajectory Scheme Robot (Base End-Effector),, 5 Polynomial 5 Polynomials (X, Y, Z 3) Sampling Time Sample Time Polynomials X,Y,Z. (X,Y,Z) Inverse Kinematics Joint Angles. Cartesian, Sampling Inverse Kinematics.., Trajectory. (X,Y, Z) sampling time Inverse Kinematics. 5 Cartesian Space polynomial X,Y,Z , (,, ) (,,)

8 . ), position x (,, ) (,,, polynomial.. 5,, (Jerk). 5, 4, 3, Jerk 2, Jerk. Velocity and Static Force Linear Velocity [A] [B], [B] Q, Q Linear Velocity [A]., Q Q [A]., Rotation Matrix R.

9 Rotational Velocity Linear Velocity, [A] origin [B] origin [A] [B] Rotation Matrix R Q, Q [B] [B] (, [B] ) [B] Q 0. [B] [A] [A] Q. Velocity Propagation from link to link Serial Mechanism robot links chain., link (,) Base link. "Propagation" link link. i+1 link link i i+1 link term. link i+1 = link i + link i+1.. link i., rotation matrix.

10 (1) link i+1. link i+1 origin link i origin link i term.., (2) (1) (2) rotation joint. prismatic. base link. base. rotational matrix base. Jacobian In Robotics Jacobian. 6 (x) (y) 6,. Vector. Y = F(X)

11 . gradient. Vector. 6x6 Matrix Jacobian. J(X). dt( ) J(X) X Y mapping. J(X) Linear Transformation.

12 Robotic. 1. Joint Space (, ) Cartesian mapping. 2. Visual Servoing(Vision Data Robot ) Image Feature Robot Joint Value mapping Jacobian multidimensional form Matrix Robotics. Robotics Jacobian. 1. v Cartesian space w Cartesian space. Jacobian Joint space, Cartesian space,,, Joint. q Joint. dt sampling time Jacobian. 1.Vector Cross Product Method 2. Differential Translation and Rotation Method 3. Strobing from the Newton-Euler Equations of Motion Jacobian Vector Cross Product Method. Vector Cross Product Method Base End-Effector,.

13 Jacobian matrix Jabobian matrix column physical. columns column joint, joint End-effecot., (q) 4 joint joints, End-effector., End-effector joint.

14 rotational joint J matrix form. Joint z position vector P. Static Forces force moment velocity link to link propagation. End-effector force force torque joint End-effector force robot static equilbrium., End-effector force structure joint force torque. End-effector

15 force static force. dynamic force.,. : link i-1 link i force : link i-1 link i torque link i force moment. link i [i] [i+1],,,. ( ) [i] [i+1] vector. static equilbrium( body) force 0, moment 0 body link force moment 0.. ( moment moment 0 ), force moment, [i] '+' [i+1] '-'..

16 [i+1] [i], (force moment ). ( ) (iteration form). joint torque joint moment.. ( joint unit vector) End-effector, joint torque simulation. Force Joint torque force Joint torque energy. energy. (dot product ) ===> (dot product ) Jacobian ( ). Cartesian Force Matrix inverse Joint space Torque. Robot Dynamics Dynamics Joint Torque.,,

17 Joint torque. Robot Dynamics Kinematics, mass, Inertia. Acceleration of a Rigid Body Rigid body. Robot dynamics. Cartesian space...,.,, Rigid body Robot Link Link., constant,..., link. Angular Acceleration() link.

18 Cartesian space (Angular Acceleration).. [B] [A], [C] [B], [A] [C]... link angular acceleration. Mass Distribution 1 rigid body mass, mass Moment of Inertia( ). Moment of Inetia Dynamics Equation. Dynamics Equation. moment of inertia matrix form inertia tensor. [A] inertia tensor.

19 Inertia tensor, inertia parallel axis theorem. inertia tensor. parallel axis theorem. Inertia tensor, inertia tensor, parallel axis theorem inertia tensor. parallel axis theorem inertia translation rotation. moment of inertia where : [A] moment of inertia, : [C] [A]. Newton-Euler Dynamics Link moment of inertia inertia tensor., inertia tensor. where (x,y,z) Link,,.

20 Dynamics Equation Biped Walking Dynamics. Newton-Euler, Lagrange-Euler, Generalized d'alembert Newton-Euler Lagrange_Euler Generalize d'alembert Multiplications O(n) O( ) O( ) Additions O(n) O( ) O( ) Kinematics Representation Rotation, Position Matrices 4x4 Homogeneous Matrices Rotation, Position Matrices Equations of motion Recursive Equations Closed-form differential Equations Closed-form differential Equations Force Balance Energy Energy 1.Newton-Euler Recursive Plant Inertial Tensors, Link Masses, Pc( Joint Link Vector), Link Rotation Matrices Joint Torque. Link Outward(0->(n-1)) Force Moment Inward(n~1) Joint Torque. 2 Planar Manipulator Newton 3., (?),. equation 2. Euler's equation rigid body(), moment N. Robot trajectory motors torques.

21 . 1.Trajectory joint (angle),,. 2.Robot Kinematics (Link, transformation matrix...) 3.mass ( link mass mass vector..) Newton-Euler force moment. joint torques. Newon-Euler formulation Outward interations i : 0 -> 5 ( propagation)

22 Inward interations i : 6 -> 1 Motor "Robocon ". 2. Torque Robot,,, robot. n 1/(n/2) robot 2., 2.., motor ( x ) moment. torque ( x ) torque, ( x ) torque. torque,..

23 torque. motor ( x ) torque. motor torque torque. torque motor torque torque. torque motor torque motor. motor torque. gear gear box. torque torque., 1/400, torque. gear. 1/400 5,. 90%, 5 1/400, 59%. 400 x 0.59 = 236. torque 239. RPM(Revolution Per Minute) 2,, motor., 1/50, motor 50., torque 1(kgf cm) 6000rpm motor torque 2(kgf cm) 3000rpm motor torque. ( torque x ) motor torque

24 ., motor,. gear, gear, motor backlash. backlash. 1 backlash, 150cm 150* = 2.61 cm.. backlash.., 2 backlash Harmonic drive ball,. backlash. wire timing belt backlash (stiffness).,, actuator. Inverse Kinematics. Inverse Kinematics., Kinematics. singular point().. actuator

25 torque.. Active Suspension.,. suspension. spring damper. spring 20cm 10cm.., spring damper actuator suspension active suspension. active suspension sensor.. motor gear air cylinder. " ", ' '....,., active suspension 2 spring damper. 'sky hook suspension'. gyro spring damper

26 . ( )..,.,,,,.,. sky hook suspension 4 6, 2.?.. mechanism... program

27 , ,..,.. projection...,,

28 ..,,.. kinematics trajectory.?.,,, m. rail,. a Newton 2 (F=ma) ma.

29 motor ma.,.,.,. " ".. "Zero Moment Point"., ZMP, sensor " " "() Zero Moment Point), "() Zero Moment Point". ZMP.,. a. ma. ma

30 . mg. N. 3 zero. 3 zero vector. mg ma. N. Q N. Q " ZMP". ZMP. "" ZMP Q Q. Q. Q. ZMP " ". Q ZMP Q.,, a., ZMP a.,, ZMP Q. ZMP., ZMP.., ZMP.

31 "

歯기구학

歯기구학 1 1.1,,.,. (solid mechanics)., (kinematics), (statics), (kinetics). ( d y n a m i c s ).,,. ( m e c h a n i s m ). ( l i n k a g e ) ( 1.1 ), (pin joint) (revolute joint) (prismatic joint) ( 1.2 ) (open