Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation Multi-body system dynamics Fa¨ ız Ben Amar amar@isir.upmc.fr ... 2014 Multi-body system dynamics

Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation 1 Introduction 2 Mathematics and Notation 3 Rotation parameters of a rigid body 4 Kinematics of rigid body 5 Kinetics of multibody system 6 Lagrange dynamics 7 Multibody system dynamics 8 Serial manipulator dynamics 9 Dynamic simulation Multi-body system dynamics

Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation Dynamics modeling and simulation Formulate the system of equations that govern the time evolution of a multibody system under the �������� action of applied (external) forces ������� These are differential-algebraic equations called ������� Equations of Motion (EOM) ������� ��������� ������� ����� � Understand how to handle various types of applied ������� ������� ���������� forces and properly include them in the EOM Understand how to compute reaction forces in any joint connecting any two bodies in the mechanism Study of equilibrium configurations, analyse their stability, linear time-frequence response Understand under what conditions a solution to the EOM exists Numerically solve the resulting (differential-algebraic) EOM Multi-body system dynamics

Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation Parametrization � � Strict or minimal (can be not representative !!) φ � � Joint (Redundant) Cartesian or absolute (highly redundant) α 2 � � � α 4 α 3 α 1 � � � � � � � θ 2 � EOM are dependent on parameters � � �� � θ 3 � � θ 1 Each differential equations is associated to one parameter � � � � �� � � � � � � �� � � � � Multi-body system dynamics

Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation Parametrization (2) Type of parametrization Strict Joint Cartesian Number of parameters Minimal Medium Important Number of differential equations Minimal Medium Important Number of algebraic equation zero Medium Important Non-linearity order High Medium Low Hard Quite hard Easy EOM obtention Computational efficiency Efficient Quite efficient Not efficient Genericity of the software Difficult Quite difficult Easy Multi-body system dynamics

Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation Typical problem in CAD of multibody systems Kinematic problem (assembly problem) Velocity and acceleration analysis Forward dynamics Inverse dynamics Static analysis of equilibrium Stability of equilibrium and linear analysis Multi-body system dynamics

Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation Forward and Inverse dynamics Forward dynamics Inverse dynamics q ( t ) q ( t ) ������� ( t ) q � ( t ) q � ( t ) ������� ( t ) τ τ �������� �������� q � � ( t ) � q � ( t ) Useful for control Useful for simulation Real time constraint and Connected with differential efficiency issue equation solver Multi-body system dynamics

Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation Vector and Matrix Geometric vector � a is a geometric vector such that � a = a x � e x + a y � e y + a z � e z Algebraic vector or matrix column a x = [ a x a y a z ] T a = a y a z Matrix a 11 a 12 . . . a 1 n a 21 a 22 . . . a 2 n A = [ a ij ] = . . . ... . . . . . . a m 1 a m 2 . . . a mn Multi-body system dynamics

Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation Dot and cross-product • Vector form : b = ab cos � a.� a,� � ( � b ) = a x b x + a y b y + a z b z b = ab sin � a × � a,� � ( � b ) � u = ( a y b z − a z b y ) � e x +( a z b x − a x b z ) � e y +( a x b y − a y b x ) � e z • Matrix form : b = a T b a.� � a × � � b = ˜ ab 0 − a z a y is the cross-product matrix. ˜ a = a z 0 − a x − a y a x 0 Multi-body system dynamics

Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation Time derivative of scalar function Derivative w.r.t time of scalar function f = f ( q 1 ( t ) , . . . , q n ( t ) , t ) dependent on parameters q ( t ) and time t dq 1 dt � ∂f dq 2 � d f ∂f ∂f + ∂f dt = . . . dt ∂q 1 ∂q 2 ∂q n ∂t . . . dq n dt ∂f d q dt + ∂f ∂t = f q ˙ = q + f t ∂ q with � ∂f � f q = ∂f ∂f ∂f ∂ q = . . . ∂q 1 ∂q 2 ∂q n Multi-body system dynamics

Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation Time derivative of vectorial function For a vectorial function f = [ f 1 , f 2 , . . . , f m ]( q 1 , q 2 , . . . , q n , t ) , its derivative w.r.t. time d f ∂ f d q dt + ∂ f = dt ∂ q ∂t = f q ˙ q + f t with f q the Jacobian of f ∂f 1 ∂f 1 ∂f 1 . . . ∂q 1 ∂q 2 ∂q n ∂f 2 ∂f 2 ∂f 2 . . . ∂q 1 ∂q 2 ∂q n ∂ f = ∂ q = f q . . . ... . . . . . . ∂f m ∂f m ∂f m . . . ∂q 1 ∂q 2 ∂q n Multi-body system dynamics

Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation Time derivative of geometric vector Consider a general vector � u How does it move in Space/Body coordinates ? What is its time derivative d� u dt ? Movement d� u depends on observer frames (space or body) ( d� u ) S = ( d� u ) B + ( d� u ) rot = ( d� u ) B + d� α × � u � d� � � d� � u u = + � ω ( B/S ) × � u dt dt S B � ω ( B/S ) is angular velocity of body frame w.r.t. space frame. Multi-body system dynamics

Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation Coordinates transformation � Let ( x, y, z ) and ( ξ, η, ζ ) 2 frames attached to ground and a body. Let � u such that ζ η � � � � u � = u x � e x + u y � e y + u z � e z � u = u ξ � e ξ + u η � e η + u ζ � e ζ � ξ We denote by u , ¯ u coordinates of � u in the ground and body frames [ u x u y u z ] T u = [ u ξ u η u ζ ] T u ¯ = u = A ¯ u with A = [ e ξ e η e ζ ] Multi-body system dynamics

Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation Properties of transformation matrix A is an rotation matrix because AA T = I 3 . Then A − 1 = A T Inverse transformation u = A T u ¯ 9 parameters of A are called direction cosines Could be used for rotation parameters ... but highly redundant Planar motion � � cos φ − sin φ s = A ¯ s = ¯ s sin φ cos φ Multi-body system dynamics

Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation Rigid body coordinates ζ η � � � � 3 position parameters ( r O ) ξ 3 orientation parameters ( A ) � � � � � � r P = r O + u P � r O + A ¯ = u P � � Multi-body system dynamics

Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation Rotation matrix � � η θ � α � � � θ θ � � � � � ξ � � � � � ∆ � � � � � � ζ ��� ��� � � v ) 2 2 sin 2 ( θ I + ˜ v sin θ + (˜ ¯ u = 2) u A ( v , θ )¯ = u Multi-body system dynamics

Outline Introduction Mathematics Rotation Kinematics Kinetics Dynamics Multibody Manipulator Simulation Euler parameters Quaternion p 1 = e 0 = cos θ Unit quaternion 2 p 2 = e 1 = v 1 sin θ 3 � p T p = ( p k ) 2 = 1 2 p 3 = e 2 = v 2 sin θ k =0 2 p 4 = e 3 = v 3 sin θ 2 Euler-Rodriguez formula becomes A = I 3 + 2˜ e ( e 0 I 3 + ˜ e ) where e = [ e 1 , e 2 , e 3 ] T Multi-body system dynamics

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