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Kane/DeAlbert dynamics for multibody system

The document discusses Kane's method for modeling multi-body systems. It begins with an introduction to multi-body systems and generalized coordinates. It then covers Kane's method which uses generalized speeds and forces to develop equations of motion in a compact form. The method encapsulates both holonomic and non-holonomic constraints. Kane's method is considered superior to other methods for modeling complex multi-body systems. The document provides details on deriving Kane's equations using virtual work principles and generalized speeds and coordinates.

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PURPOSE: DYNAMICS OF MULTI BODY SYSTEM BY KANE’S METHOD FOCUS: Introduction to multibody system Dynamic Modeling of Multibody system Kane’s Method of Multi Body System Application of Kane’s Method Equation 1 KANE DYNAMICS WITH APPLICATION TO MULTI BODY SYSTEM LIKE IN MECHANISMS AND MANIPULATORS preparedbyYohannesR.@JiT
 In Systems where all parts are not best described in an inertial or "world" reference frame are referred to as multi-body systems (MBSs).  These are common in: Robotics Aerospace Aviation Industrial Automation 2 Introduction to multibody system preparedbyYohannesR.@JiT
 To model a given dynamic system S, one has choose variables to describe the configuration of S, or variables that specify the location of a reference point and orientation of a reference frame fixed within each body of S. These variables named as configuration variables.  For mechanical scenario, a multibody system is a system that consists a number of rigid bodies (referred to as links) connected in succession by kinematic pairs (referred to as joints). Base Link0 Joint1 Link2 Link3Joint3 End of Arm Link1 Joint2 3 preparedbyYohannesR.@JiT
4  Human hands can be act as MBS in which the arrangement of bones(link) and kinematic pair of joints provides dexterity (manipulating objects)  Each joint represent a degree of freedom; there are 27bones, 22joints and thus 22DoF Example of MBS preparedbyYohannesR.@JiT
5 The basic laws of dynamics can be formulated (expressed mathematically) in several ways other that that given by Newton’s Laws. The most important are: (a) D’Alembert Principle (b) Lagrange’s Equations (c) Hamilton’s Equations (f) Kane’s Equations (d) Gibbs-Appell’s Equations Popular Methods for Modeling of Multi-body Systems  According to literatures there are two classes of methods to model Multi-body Systems  Vector Methods Newton-Euler equation, Kane's equation , D’Alembert Principle  Scalar Methods Lagrangian Dynamics equation, Gibbs-Appell’s Equations Kane's method borrows concepts from both, but is classified as a Vector Method. Can be named optimized equation (e) Newton-Euler’s Equations Dynamic Modeling of Multibody system preparedbyYohannesR.@JiT
6  Kane’s method (originally called Lagrange form of D'Alambert's principle) which is a powerful tool for developing dynamical equations for MBS motion  Applying the Newton-Euler method requires that force and moment balances be applied for each body taking in consideration every interactive and constraint force.  Therefore, the method is inefficient when only a few of the system’s forces need to be solved for.  The major disadvantage of Lagrange’s Equations method is the need to differentiate scalar energy functions (kinetic and potential energy).  In Kane’s method, With the use of generalized forces the need for examining interactive and constraint forces between bodies is eliminated  (Huston 1990) argue that; Kane’s method provides combined means to develop the dynamics equations for multibody systems that lends itself to automated numerical computation. preparedbyYohannesR.@JiT  Essentially all methods for obtaining equations of motion are equivalent.  However, some are more suited for multibody dynamics than others.
 For example, if a revolute joint connects two bodies, only the joint angle is needed to describe the configuration of the second body, if the configuration of the first is known.  Compared to a set of variables that specifies the location and orientation of each body relative to a common ground, a set of generalized coordinates is reduced in number i.e., n<6v, where v is the number of bodies of S and spatial motion is being considered.  If M is the number of configuration constraints encapsulated by the choice of generalized coordinates, then n = 6v-M.  The generalized coordinates describe only the allowed configurations and thus encapsulate certain configuration constraints. 7 preparedbyYohannesR.@JiT  Generalized coordinates are a set of convenient coordinates, usually independent of one another, used to describe a particular configuration of a system
Kane's method is touted as a superior approach by it proponents because it:  Encapsulates holonomic (position) constraints by the use of generalized coordinates (as in the Lagrangian method).  Also encapsulates non- holonomic (velocity) constraints through the use of generalized speeds. (Which requires Lagrange's Method of Undetermined Multipliers)  Results in a compact, first order representation of the equations of motion. (ODE)  Is more systematic and therefore easier to learn .  Is becoming the industry standard where complex systems need to be modeled. Why Kane's Method? 8 preparedbyYohannesR.@JiT
9 Kane’s Equations Consider an open-chain multibody system of N interconnected rigid bodies each subject to external and constraint forces these external forces can be transformed into an equivalent force and torque Fk and Mk passing through Gk ,which is the mass center of the body k , (k = 1,2…N). Similar to the external forces, the constraint forces may be written as and Using D'Alambert's principle for the force equilibrium of body k, the following is obtained 0  c kkk FFF  kkk amF   where is the inertia force of body k. The concept of virtual work may be described as follows for a system of N particles with N degrees of freedom Professor Thomas R. Kane 1924 - Stanford University Kane’s Equations Using the Principle of Virtual Work c kF  c kM  Kane’s Method of Multi Body System preparedbyYohannesR.@JiT
10 The virtual work is then defined as:   N i ii rFW 1   Where is the resultant force acting on the ith particle and is the position vector of the particle in the inertial reference frame. is the virtual displacement, which is imaginary in the sense that it is assumed to occur without the passage of time. iF  ir ir Now applying the concept of virtual work to our multibody system considering only the work due to the forces on the system we obtain: 0)(   k c kkk rFFFW   (k = 1,2,…,N) The constraints that are commonly encountered are known as workless constraints so… 0 k c k rF   Which simplifies the virtual work equation to: 0)(   kkki rFFW   The positions vector may also be written as: ),( tqrr rkk   t r dt dq q r r kr r k k         t r q q r k r r k           preparedbyYohannesR.@JiT
11 Taking the partial derivative of with respect to we obtain kr rq r k r k q r q r         r k r k q r q v         or Since the virtual displacement is arbitrary without violating the constraints we can write * as: rq 0  KK ff Kf  Kf generalized active generalized inertia forces r k kK q v Ff      r k kK q v Ff       In a similar fashion it can be shown using virtual work that the moments can be written as 0  KK MM KM generalized active moments  KM generalized inertia moments r k kK q TM       r k kkkK q IIM         )( By superposition of the force and moment equations we arrive at Kane’s equations: 0  kk FF KKK MfF    KKK MfF preparedbyYohannesR.@JiT
12  tri  ir   tvi   tPi tP i'  ttP i ' ir   Virtual Path True Path (P) Newtonian or Dynamic Path The Constraint Space at t mjra N i i j i ,,10 1                        0 0 0 21 21 21 tttt trtr trtr ii ii    1t 2t Kane’s Equations can be also derived in terms of generalized coordinates we can write: Kane’s Equations Using the Generalized Coordinates system           Niktqzjtqyitqxtqqrtqr iiinii .,1,,,,,,, 1      1   Nikdzjdyidxdt t r dq q r tqrd iii i n j j j i i .,1, 1            2 preparedbyYohannesR.@JiT
13   Ni t r q q r td tqrd rv i n j j j ii ii .,1 , 1                3 Kane and Levinson have shown that with the n generalized coordinates , is useful to define another n variables , which are linear functions of the n : jq iu jq nrZqYu r n j jrjr .,1 1    4 where the matrix is invertible and    nj nrrjYY ,1 ,1          nj nrrjWWY ,1 ,1 1     njXuWq j n r rrjj .,1 1    5 jrrjrj XandZWY ,, are functions of tandq are called Generalized Speed (also Nonholonomic Velocities, Quasivelocities. etc.) and are not unique. iu preparedbyYohannesR.@JiT where the elements of the (nxn) matrix Y and (nx1) matrix Z are functions of qi (i = 1; …….;n) and possibly the time t. Reciprocal relations express the generalized coordinate derivatives in terms of the generalized speeds:
14 Kane’s Equations (continue) Non holonomic constraints are linear relations among either or the ; for m Non holonomic constraints: 6 7 where k may be n-m or n, depending on whether the non holonomic constraints are incorporated. iu jq nmnsBuAu s mn r rsrs .,1 1      If we substitute equations (6) in (5) we obtain a more general expression for : jq njXuWq j k r rrjj .,1 1    Let substitute equation (6) in (3):   Ni t r X q r uW q r t r XuW q r td tqrd rv i k r n j j j i r n j rj j i i n j j k r rrj j ii ii .,1, , 1 11 1 1                                             From this equation we can see that       n j rj j i r i W q r u v 1  preparedbyYohannesR.@JiT njXuWq j n r rrjj .,1, 1     Eq.-7 represents for kinematical differential equations and form the first part of the state equations (equations of motion)
15 Kane’s Equations (continue) 8 9 Let use now the equation of differential work: By defining we obtain: t r X q r v i n j j j ii t          1 Nivu u v v t i k r r r i i .,1 1             N i iii N i ii rdamrdFdW 11  Equation (9) is now rewritten using (8). On the left side we obtain: dtvu u v FdtvFrdF N i t i k r r r i i N i ii N i ii                       1 111   10 Similarly, the right side of (9) becomes: dtvu u v amdtvamrdam N i t i k r r r i ii N i iii N i iii                       1 111    11 Equations (10) and (11) are equated and terms re collected:     0 11 11                               dtamFvdtuam u v F u v N i iii t i k r r N i ii r i N i i r i   12 preparedbyYohannesR.@JiT
16 Kane’s Equations (continue)     0 11 11                               dtamFvdtuam u v F u v N i iii t i k r r N i ii r i N i i r i   12 The and dt are nonzero and independent and so the coefficients of each of them must be zero. Also using Newton’s Second Law: krur ,1 0 iii amF  nrZqYu r n j jrjr ,.,1 1    krF u v F N i i r i r ,,1 1          kram u v F N i ii r i r ,,1' 1         krFF rr ,,10'  4 13 14 15 Generalized Speeds Generalized Active Forces Generalized Inertia Forces summary of Kane’s Equations we have confined Kane's equation of motion: preparedbyYohannesR.@JiT 0  kk FF Similar to VRP
17 Application of Kane’s Method Equation Problem 1 The first problem is a spring-mass-pendulum problem with frictionless sliding. This was problem is good for showing the general procedure of Kane’s method. Step 1) Define important points as the center of mass of A and particle P general procedure for developing Kane's equations of motion Solution Step 2) Select generalized coordinates as shown in the figure and generate velocity and acceleration expressions for the important points. 11 ˆnuv AN   APPNANPN rvv    )ˆ()ˆ(ˆ 23211 bLbunu  1211 ˆˆ bLunu  11 ˆnuaAN    2 2 21211 ˆˆˆ bLubLunuaPN    Step 3) Construct a partial velocity table. preparedbyYohannesR.@JiT
18 0  kk FF Step 4) apply the concept of virtual work P r NA r N k vngMvngMnKqF   )ˆ()ˆˆ( 222111 11 KqF  )sin( 222 qLgMF  P r NA r NAN k vngMvaMF   )ˆ()( 221 Generalized Active Forces Generalized inertia Forces ))sin()cos(( 2 2 22212111 qLuqLuuMuMF   ))cos(( 2 22122 LuqLuMF   Step 5) Assemble the equation and form matrix for final solution unknown variables 0  rr FF                      )sin( )sin( )cos( )cos()( 22 12 2 22 2 1 2 222 2221 qLgM KqqLuM u u LMqLM qLMMM   preparedbyYohannesR.@JiT
Manipulator dynamic model 19  As stated by Angeles et al., 1989, Kane’s equations sometimes referred to as D’Alambert’s equations in Lagrangian form which are powerful in robot manipulator dynamics preparedbyYohannesR.@JiT
20 Problem 2 This problem is useful in that it shows how auxiliary generalized speeds can be introduced to bring constraint forces and torques into evidence. In this case we will introduce u3 to find an expression for Tc (constraint torque about ) The joints at O and P are revolute. Body A and B are uniform rods with length 4L and 2L respectively. Body A has two times the mass of body B. 1 ˆa preparedbyYohannesR.@JiT
21 Step 1) Choose important points: Center of Mass of bodies A and B, and point P. Step 2) Select generalized coordinates as shown in the figure (plus auxiliary generalized coordinate u3) and generate velocity and acceleration expressions for the important points. The prime ( *) in the equations below indicates that the specified quantities contain the auxiliary generalized coordinate. Body A 2113 ˆˆ auauAN   )ˆˆ(2 1123 auauLrvv OAANONAN     21 ˆauAN    3 2 111 ˆ2ˆ2 aLuauLa AN    Point P )ˆˆ(4 1123 auauLrvv OPANONPN    preparedbyYohannesR.@JiT
22 Body B 322113 ˆ)ˆˆ( auauauBAANBN     3212322231221 ˆ)(ˆ)4(ˆ)4( asucuasuuacuuLrvv PBBNPNBN     1213221 ˆˆˆ auuauauBAANBAANBN     )(   PBBNBNPBBNPNBN rraa      322122 2 212 2 22 2 13212212221222 2 111 ˆˆˆ)(ˆ)(ˆˆˆˆ4 acuuacuasusuLasucuuasuacuLauau   Step 3) Construct a partial velocity table. preparedbyYohannesR.@JiT
23 0  kk FF B r N A r N c A r N k vnmgaTvnmgF        )ˆ()ˆ()ˆ2( 333  )8( 1211 cssmgLF  212 cmgLsF  cTcmgLcF  213 Step 4) apply the concept of virtual work A r NANA A ANA A AN A r NAN k IIvamF                )2( B r NBNB B BNB B AN B r NBN IIvam                )( 22 2 11 22 2 ˆˆ 3 4 ˆˆ 3 4 000 0 3 4 0 00 3 4 2 aaLaaLL L mI A A                     33 2 11 2 2 2 ˆˆ 3 1ˆˆ 3 1 3 1 00 000 00 3 1 bbLbbL L L mI B B                   preparedbyYohannesR.@JiT
24 )( 3 )8( 3 8 2221 2 21 2 1 2 1 2 1 csuusu mL umLu mL F                )2())((4)( 212212 2 2 2 12221 2 sucuusuuscuumL    )4())(()( 3 2 222 2 122 2 2 2 12221 22 1222 2 2 usuuscuuscuumLucsu mL F          )2()4(4)2( 3 212212 2 222 2 1 22 221221 2 3 sucuucusuumLcuucsu mL F                                                          2 2 2 21 2 2 2 12 2 1 22 221 2 2 2 2 12 2 21 2 2 2 122 2 22 2 1 2 1212122 2 2 2 12 2 2221 2 2 1 2 2 2 2 22 2 2 2 2 2 2 22 2 2 2 2 16216 3 2 )4()( 3 )8()2)(4( 3 14 3 0 3 04 3 4 12 3 8 umLuucmLcmglcumLcuu mL uusmLcmglsuucsmLcsu mL cssmgluucsuusmLcsuu mL T u u smLsmLcs mL mL mL cmL cmLs mL mL mL c   Step 5) Assemble the equation and form matrix for final solution unknown variables preparedbyYohannesR.@JiT
25 preparedbyYohannesR.@JiT Velocity propagation to find unknown linear and angular velocities of 2R Problem 3
26 preparedbyYohannesR.@JiT
27 preparedbyYohannesR.@JiT or
28 preparedbyYohannesR.@JiT
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Kane/DeAlbert dynamics for multibody system

  • 1.
    PURPOSE: DYNAMICS OFMULTI BODY SYSTEM BY KANE’S METHOD FOCUS: Introduction to multibody system Dynamic Modeling of Multibody system Kane’s Method of Multi Body System Application of Kane’s Method Equation 1 KANE DYNAMICS WITH APPLICATION TO MULTI BODY SYSTEM LIKE IN MECHANISMS AND MANIPULATORS preparedbyYohannesR.@JiT
  • 2.
     In Systemswhere all parts are not best described in an inertial or "world" reference frame are referred to as multi-body systems (MBSs).  These are common in: Robotics Aerospace Aviation Industrial Automation 2 Introduction to multibody system preparedbyYohannesR.@JiT
  • 3.
     To modela given dynamic system S, one has choose variables to describe the configuration of S, or variables that specify the location of a reference point and orientation of a reference frame fixed within each body of S. These variables named as configuration variables.  For mechanical scenario, a multibody system is a system that consists a number of rigid bodies (referred to as links) connected in succession by kinematic pairs (referred to as joints). Base Link0 Joint1 Link2 Link3Joint3 End of Arm Link1 Joint2 3 preparedbyYohannesR.@JiT
  • 4.
    4  Human handscan be act as MBS in which the arrangement of bones(link) and kinematic pair of joints provides dexterity (manipulating objects)  Each joint represent a degree of freedom; there are 27bones, 22joints and thus 22DoF Example of MBS preparedbyYohannesR.@JiT
  • 5.
    5 The basic lawsof dynamics can be formulated (expressed mathematically) in several ways other that that given by Newton’s Laws. The most important are: (a) D’Alembert Principle (b) Lagrange’s Equations (c) Hamilton’s Equations (f) Kane’s Equations (d) Gibbs-Appell’s Equations Popular Methods for Modeling of Multi-body Systems  According to literatures there are two classes of methods to model Multi-body Systems  Vector Methods Newton-Euler equation, Kane's equation , D’Alembert Principle  Scalar Methods Lagrangian Dynamics equation, Gibbs-Appell’s Equations Kane's method borrows concepts from both, but is classified as a Vector Method. Can be named optimized equation (e) Newton-Euler’s Equations Dynamic Modeling of Multibody system preparedbyYohannesR.@JiT
  • 6.
    6  Kane’s method(originally called Lagrange form of D'Alambert's principle) which is a powerful tool for developing dynamical equations for MBS motion  Applying the Newton-Euler method requires that force and moment balances be applied for each body taking in consideration every interactive and constraint force.  Therefore, the method is inefficient when only a few of the system’s forces need to be solved for.  The major disadvantage of Lagrange’s Equations method is the need to differentiate scalar energy functions (kinetic and potential energy).  In Kane’s method, With the use of generalized forces the need for examining interactive and constraint forces between bodies is eliminated  (Huston 1990) argue that; Kane’s method provides combined means to develop the dynamics equations for multibody systems that lends itself to automated numerical computation. preparedbyYohannesR.@JiT  Essentially all methods for obtaining equations of motion are equivalent.  However, some are more suited for multibody dynamics than others.
  • 7.
     For example,if a revolute joint connects two bodies, only the joint angle is needed to describe the configuration of the second body, if the configuration of the first is known.  Compared to a set of variables that specifies the location and orientation of each body relative to a common ground, a set of generalized coordinates is reduced in number i.e., n<6v, where v is the number of bodies of S and spatial motion is being considered.  If M is the number of configuration constraints encapsulated by the choice of generalized coordinates, then n = 6v-M.  The generalized coordinates describe only the allowed configurations and thus encapsulate certain configuration constraints. 7 preparedbyYohannesR.@JiT  Generalized coordinates are a set of convenient coordinates, usually independent of one another, used to describe a particular configuration of a system
  • 8.
    Kane's method istouted as a superior approach by it proponents because it:  Encapsulates holonomic (position) constraints by the use of generalized coordinates (as in the Lagrangian method).  Also encapsulates non- holonomic (velocity) constraints through the use of generalized speeds. (Which requires Lagrange's Method of Undetermined Multipliers)  Results in a compact, first order representation of the equations of motion. (ODE)  Is more systematic and therefore easier to learn .  Is becoming the industry standard where complex systems need to be modeled. Why Kane's Method? 8 preparedbyYohannesR.@JiT
  • 9.
    9 Kane’s Equations Consider anopen-chain multibody system of N interconnected rigid bodies each subject to external and constraint forces these external forces can be transformed into an equivalent force and torque Fk and Mk passing through Gk ,which is the mass center of the body k , (k = 1,2…N). Similar to the external forces, the constraint forces may be written as and Using D'Alambert's principle for the force equilibrium of body k, the following is obtained 0  c kkk FFF  kkk amF   where is the inertia force of body k. The concept of virtual work may be described as follows for a system of N particles with N degrees of freedom Professor Thomas R. Kane 1924 - Stanford University Kane’s Equations Using the Principle of Virtual Work c kF  c kM  Kane’s Method of Multi Body System preparedbyYohannesR.@JiT
  • 10.
    10 The virtual workis then defined as:   N i ii rFW 1   Where is the resultant force acting on the ith particle and is the position vector of the particle in the inertial reference frame. is the virtual displacement, which is imaginary in the sense that it is assumed to occur without the passage of time. iF  ir ir Now applying the concept of virtual work to our multibody system considering only the work due to the forces on the system we obtain: 0)(   k c kkk rFFFW   (k = 1,2,…,N) The constraints that are commonly encountered are known as workless constraints so… 0 k c k rF   Which simplifies the virtual work equation to: 0)(   kkki rFFW   The positions vector may also be written as: ),( tqrr rkk   t r dt dq q r r kr r k k         t r q q r k r r k           preparedbyYohannesR.@JiT
  • 11.
    11 Taking the partialderivative of with respect to we obtain kr rq r k r k q r q r         r k r k q r q v         or Since the virtual displacement is arbitrary without violating the constraints we can write * as: rq 0  KK ff Kf  Kf generalized active generalized inertia forces r k kK q v Ff      r k kK q v Ff       In a similar fashion it can be shown using virtual work that the moments can be written as 0  KK MM KM generalized active moments  KM generalized inertia moments r k kK q TM       r k kkkK q IIM         )( By superposition of the force and moment equations we arrive at Kane’s equations: 0  kk FF KKK MfF    KKK MfF preparedbyYohannesR.@JiT
  • 12.
    12  tri  ir   tvi   tPitP i'  ttP i ' ir   Virtual Path True Path (P) Newtonian or Dynamic Path The Constraint Space at t mjra N i i j i ,,10 1                        0 0 0 21 21 21 tttt trtr trtr ii ii    1t 2t Kane’s Equations can be also derived in terms of generalized coordinates we can write: Kane’s Equations Using the Generalized Coordinates system           Niktqzjtqyitqxtqqrtqr iiinii .,1,,,,,,, 1      1   Nikdzjdyidxdt t r dq q r tqrd iii i n j j j i i .,1, 1            2 preparedbyYohannesR.@JiT
  • 13.
    13   Ni t r q q r td tqrd rv i n j j j ii ii.,1 , 1                3 Kane and Levinson have shown that with the n generalized coordinates , is useful to define another n variables , which are linear functions of the n : jq iu jq nrZqYu r n j jrjr .,1 1    4 where the matrix is invertible and    nj nrrjYY ,1 ,1          nj nrrjWWY ,1 ,1 1     njXuWq j n r rrjj .,1 1    5 jrrjrj XandZWY ,, are functions of tandq are called Generalized Speed (also Nonholonomic Velocities, Quasivelocities. etc.) and are not unique. iu preparedbyYohannesR.@JiT where the elements of the (nxn) matrix Y and (nx1) matrix Z are functions of qi (i = 1; …….;n) and possibly the time t. Reciprocal relations express the generalized coordinate derivatives in terms of the generalized speeds:
  • 14.
    14 Kane’s Equations (continue) Nonholonomic constraints are linear relations among either or the ; for m Non holonomic constraints: 6 7 where k may be n-m or n, depending on whether the non holonomic constraints are incorporated. iu jq nmnsBuAu s mn r rsrs .,1 1      If we substitute equations (6) in (5) we obtain a more general expression for : jq njXuWq j k r rrjj .,1 1    Let substitute equation (6) in (3):   Ni t r X q r uW q r t r XuW q r td tqrd rv i k r n j j j i r n j rj j i i n j j k r rrj j ii ii .,1, , 1 11 1 1                                             From this equation we can see that       n j rj j i r i W q r u v 1  preparedbyYohannesR.@JiT njXuWq j n r rrjj .,1, 1     Eq.-7 represents for kinematical differential equations and form the first part of the state equations (equations of motion)
  • 15.
    15 Kane’s Equations (continue) 8 9 Letuse now the equation of differential work: By defining we obtain: t r X q r v i n j j j ii t          1 Nivu u v v t i k r r r i i .,1 1             N i iii N i ii rdamrdFdW 11  Equation (9) is now rewritten using (8). On the left side we obtain: dtvu u v FdtvFrdF N i t i k r r r i i N i ii N i ii                       1 111   10 Similarly, the right side of (9) becomes: dtvu u v amdtvamrdam N i t i k r r r i ii N i iii N i iii                       1 111    11 Equations (10) and (11) are equated and terms re collected:     0 11 11                               dtamFvdtuam u v F u v N i iii t i k r r N i ii r i N i i r i   12 preparedbyYohannesR.@JiT
  • 16.
    16 Kane’s Equations (continue)    0 11 11                               dtamFvdtuam u v F u v N i iii t i k r r N i ii r i N i i r i   12 The and dt are nonzero and independent and so the coefficients of each of them must be zero. Also using Newton’s Second Law: krur ,1 0 iii amF  nrZqYu r n j jrjr ,.,1 1    krF u v F N i i r i r ,,1 1          kram u v F N i ii r i r ,,1' 1         krFF rr ,,10'  4 13 14 15 Generalized Speeds Generalized Active Forces Generalized Inertia Forces summary of Kane’s Equations we have confined Kane's equation of motion: preparedbyYohannesR.@JiT 0  kk FF Similar to VRP
  • 17.
    17 Application of Kane’sMethod Equation Problem 1 The first problem is a spring-mass-pendulum problem with frictionless sliding. This was problem is good for showing the general procedure of Kane’s method. Step 1) Define important points as the center of mass of A and particle P general procedure for developing Kane's equations of motion Solution Step 2) Select generalized coordinates as shown in the figure and generate velocity and acceleration expressions for the important points. 11 ˆnuv AN   APPNANPN rvv    )ˆ()ˆ(ˆ 23211 bLbunu  1211 ˆˆ bLunu  11 ˆnuaAN    2 2 21211 ˆˆˆ bLubLunuaPN    Step 3) Construct a partial velocity table. preparedbyYohannesR.@JiT
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    18 0  kk FF Step4) apply the concept of virtual work P r NA r N k vngMvngMnKqF   )ˆ()ˆˆ( 222111 11 KqF  )sin( 222 qLgMF  P r NA r NAN k vngMvaMF   )ˆ()( 221 Generalized Active Forces Generalized inertia Forces ))sin()cos(( 2 2 22212111 qLuqLuuMuMF   ))cos(( 2 22122 LuqLuMF   Step 5) Assemble the equation and form matrix for final solution unknown variables 0  rr FF                      )sin( )sin( )cos( )cos()( 22 12 2 22 2 1 2 222 2221 qLgM KqqLuM u u LMqLM qLMMM   preparedbyYohannesR.@JiT
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    Manipulator dynamic model 19 As stated by Angeles et al., 1989, Kane’s equations sometimes referred to as D’Alambert’s equations in Lagrangian form which are powerful in robot manipulator dynamics preparedbyYohannesR.@JiT
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    20 Problem 2 This problemis useful in that it shows how auxiliary generalized speeds can be introduced to bring constraint forces and torques into evidence. In this case we will introduce u3 to find an expression for Tc (constraint torque about ) The joints at O and P are revolute. Body A and B are uniform rods with length 4L and 2L respectively. Body A has two times the mass of body B. 1 ˆa preparedbyYohannesR.@JiT
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    21 Step 1) Chooseimportant points: Center of Mass of bodies A and B, and point P. Step 2) Select generalized coordinates as shown in the figure (plus auxiliary generalized coordinate u3) and generate velocity and acceleration expressions for the important points. The prime ( *) in the equations below indicates that the specified quantities contain the auxiliary generalized coordinate. Body A 2113 ˆˆ auauAN   )ˆˆ(2 1123 auauLrvv OAANONAN     21 ˆauAN    3 2 111 ˆ2ˆ2 aLuauLa AN    Point P )ˆˆ(4 1123 auauLrvv OPANONPN    preparedbyYohannesR.@JiT
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    22 Body B 322113 ˆ)ˆˆ( auauauBAANBN    3212322231221 ˆ)(ˆ)4(ˆ)4( asucuasuuacuuLrvv PBBNPNBN     1213221 ˆˆˆ auuauauBAANBAANBN     )(   PBBNBNPBBNPNBN rraa      322122 2 212 2 22 2 13212212221222 2 111 ˆˆˆ)(ˆ)(ˆˆˆˆ4 acuuacuasusuLasucuuasuacuLauau   Step 3) Construct a partial velocity table. preparedbyYohannesR.@JiT
  • 23.
    23 0  kk FF B r N A r N c A r N kvnmgaTvnmgF        )ˆ()ˆ()ˆ2( 333  )8( 1211 cssmgLF  212 cmgLsF  cTcmgLcF  213 Step 4) apply the concept of virtual work A r NANA A ANA A AN A r NAN k IIvamF                )2( B r NBNB B BNB B AN B r NBN IIvam                )( 22 2 11 22 2 ˆˆ 3 4 ˆˆ 3 4 000 0 3 4 0 00 3 4 2 aaLaaLL L mI A A                     33 2 11 2 2 2 ˆˆ 3 1ˆˆ 3 1 3 1 00 000 00 3 1 bbLbbL L L mI B B                   preparedbyYohannesR.@JiT
  • 24.
    24 )( 3 )8( 3 8 2221 2 21 2 1 2 1 2 1 csuusu mL umLu mL F               )2())((4)( 212212 2 2 2 12221 2 sucuusuuscuumL    )4())(()( 3 2 222 2 122 2 2 2 12221 22 1222 2 2 usuuscuuscuumLucsu mL F          )2()4(4)2( 3 212212 2 222 2 1 22 221221 2 3 sucuucusuumLcuucsu mL F                                                          2 2 2 21 2 2 2 12 2 1 22 221 2 2 2 2 12 2 21 2 2 2 122 2 22 2 1 2 1212122 2 2 2 12 2 2221 2 2 1 2 2 2 2 22 2 2 2 2 2 2 22 2 2 2 2 16216 3 2 )4()( 3 )8()2)(4( 3 14 3 0 3 04 3 4 12 3 8 umLuucmLcmglcumLcuu mL uusmLcmglsuucsmLcsu mL cssmgluucsuusmLcsuu mL T u u smLsmLcs mL mL mL cmL cmLs mL mL mL c   Step 5) Assemble the equation and form matrix for final solution unknown variables preparedbyYohannesR.@JiT
  • 25.
    25 preparedbyYohannesR.@JiT Velocity propagation tofind unknown linear and angular velocities of 2R Problem 3
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    31 10Q 4 urattention preparedbyYohannesR.@JiT