1 / 87

CHAPTER 4

CHAPTER 4. Robot Dynamics. •. ( q , q ). L = K – P. Potential Energy. Lagrangian. Kinetic Energy. Dynamics. Joint Forces/Torque  Joint Motion (End Effector Motion). LAGRANGIAN MECHANICS APPROACH

doctor
Télécharger la présentation

CHAPTER 4

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. CHAPTER 4 Robot Dynamics

  2. • (q, q) L = K – P Potential Energy Lagrangian Kinetic Energy Dynamics Joint Forces/Torque  Joint Motion (End Effector Motion) LAGRANGIAN MECHANICS APPROACH – “Energy-based” approach to dynamics The Lagrangian & energy terms can be expressed in any convenient coordinate system (that will simplify the problem)

  3. • where qi are the coordinates in which the kinetic & potential energy are expressed. qi = velocity Fi = corresponding Force torque in direction of qi generalized coordinates closed form solution • Dynamics The dynamics equations, in terms of the coordinates used to express the Kinetic & Potential Energy, are: Ref: B.J. Torby, “Advanced Dynamics for Engineers”, Halt Rinehart, 1984.

  4. m (pt mass) • • • L K = ½ mv2 = ½ m(L)2 = ½ mL22 P = mg Lsin L = K – P • T • L = ½ mL22 – mg Lsin = ½ mL2(2) = mL2 = -mg Lcos • • • Dynamics Ex: 1 - DOF Robot

  5. • •• T = mL2 + mg Lcos Dynamics T = Torque (in direction of ) Note: This can be derived from Fundamental laws of motion e.g. Newton’s eq. F = ma T = I leading to Newton-Euler formulation - recursive solution

  6. ir Ti {i} {o} Base • • Note that ir is constant r = velocity of point • Kinetic Energy Velocity of a Point on the manipulator: given a point ir described with respect to link i, its position in base coords is r = Ti ir

  7. • • • • • || r ||22 = r• r = Trace (r rT) • • • • Kinetic Energy

  8. • • • Kinetic Energy of link i • • Ji = pseudo inertia matrix Kinetic Energy Kinetic Energy of a particle of mass dm located on link i at ir is

  9. Kinetic Energy where all integrals are ∫link i and rT = [ ix iy iz 1 ]T Symmetric matrix

  10. Recall That Ixx = ∫(y2 + z2) dm Iyy = ∫(x2 + z2) dm Izz = ∫(x2 + y2) dm Ixy = ∫ xy dm Ixz = ∫ xz dm Iyz = ∫ yz dm mx = ∫ x dm my = ∫ y dm mz = ∫ z dm Kinetic Energy

  11. Then Kinetic Energy

  12. • • Adding effect of actuators Kactuatori = ½ Iaiqi2 Iai = actuator inertia, rotational joint i = equivalent mass, translational joint i • Kinetic Energy The total kinetic energy of an N-joint manipulator is then neglecting actuator inertia

  13. P = -mg• r ⊝ because g is opposite height above some reference zero r = position of center of mass of object g = gravity force vector Potential Energy

  14. Ex: a mass m at r = 10i + 20j + 30k in a gravity field g = 0i + 0j + -32.2k has a P = 966 m The potential energy of a link whose center of mass is described by iri with respect to link i coord frame Ti is then Pi = -migT Tiiri reference is base frame where g = [ gx gy gz 0 ]T in base frame coords Total Potential Energy is Potential Energy

  15. • • • Lagrangian Lagrangian L = K – P

  16. • • • • change to k Dynamic Equations

  17. • • • since for p > i, we have • • • Note that

  18. •• •• • • • • used Trace (M) = Trace(MT)

  19. • • • •

  20. • • Interchanging dummy indices of summation j & k in 2nd term, & then combining with the first,

  21. Therefore, • •• •• • •

  22. •• •• • • OR Exchanging dummy summation indices p and i for i and j,

  23. Model of Robot Dynamics •• •• • • Inertial Centripetal + Coriolis Gravity

  24. OR F = D(q) q + C(q, q) + G(q) matrix-vector form •• • (Kinetic Energy) (Potential Energy) • • Note the diff indices ( p = max(i, j) ) compared to previous eqn for Fi can be verified to be correct

  25. Simplifying the Dynamic Model Notation: Define 

  26. Rotational joint Translational joint where

  27. Summary

  28. In general F = D(q) q + C(q, q) + G(q) + b(q, q) where actuator inertia Iai are lumped into D(q) : dii dii + Iai b(q, q): frictional force/torque vector (damping + coulomb friction) OR F = D(q) q + H(q, q) H(q, q) = coupling vector that incorporates C(q, q), G(q) & b(q, q) •• • • • •• • • • • A more realistic dynamic model

  29. relates Applied Joint Torque with Actuator Inputs U = [ Ui(t) ] (Voltage) L R Effective inertia + + V Vb Jeff _ _ m m Beff Effective viscous Coeff. Of friction (Damping) Ref: 1. Fu, Lee Gonzalez 2. Spong & Vidyasagar Actuator Dynamics

  30. Gear Ratio: N m = N L m = N L m = N L L referred to motor shaft = L* L* = L JL • • Jm •• •• L L BL m m Bm •• • m = (Jmm + Bmm) + L Mechanical Eq. Actuator Dynamics Torque from = Torque on + Torque on Load motor shaft motor Referred to motor shaft

  31. where Vb = KEm (Back Emf) KE in (Voltage constant of motor) m = KT i (torque delivered by motor) KT in (Torque constant of motor) (Note that if KE is in and KT is in , they are numerically equal, i.e. KE = KT) Actuator Dynamics Electrical Eqns:

  32. •• • • m   _ m m V(s) I KT + Vb KE Actuator Dynamics Summary:

  33. For the case when load is just an inertia load: JL, L = JLL + BLL then •• • • •• • •• •• • Jeff Beff •• • m = Jeff m + Beff m on motor side where Actuator Dynamics

  34. • m L m m V(s) I + KT _  KE Actuator Dynamics

  35. _ m V   + _ + KE • • Actuator Dynamics For Electro – Mechanical Systems, it is assumed that the electrical time constant L/R is much smaller than the Mechanical time constant Jm/Bm Setting L/R = 0 results in

  36. •• • • • • Torque delivered on motor side Actuator Dynamics The robot dynamics equations incorporating actuator dynamics (ignoring L), in terms of robot (side) joint coords are:

  37. • • •• • Actuator Dynamics Note Jm, Bm, KE, KT, N, R are different for each actuator-drive system

  38. Summary of Properties & Characteristics of Inertia Coefficients • Inertia Matrix • Symmetric : dij = dji for all i and j • Positive definite : dii > 0 for all i • dij2 < diidjj for all i  j • Self Inertia Coefficient (i = j) • joint i translational: • dii = mi + … + mN = constant • joint i rotational: • dii = (mi + … + mN)Kii2 = f(qi+1, … , qN)

  39. Coeff of Coupling Summary of Properties & Characteristics of Inertia Coefficients Mutual Inertia Coefficients (for j > i) i = Transl, j = Transl: dij = (mj + … + mN) (zi-1• zj-1) = f(qi+1, … , qj-1) i = Transl, j = Rot: dij = (mj + … + mN) (zi-1x zj-1) • pj = f(qi+1, … , qN) i = Rot, j = Transl: dij = (mj + … + mN) (zj-1x zi-1) • pi = f(qi+1, … , qj-1, qj+1, … , qN) i = Rot, j = Rot: dij = (mj + … + mN) kij = f(qi+1, … , qN)

  40. Centrifugal And Coriolis Coefficients • Cijm (Centrifugal & Coriolis coefficients) are • related to the inertial matrix through the • christoffel symbols: • These Christoffel symbols coupled with the • functional dependence of the inertial coef: • dij = f(qs+1, … , qN) where s = min(i, j) • simplify the formulation of the Centrifugal & • Coriolis coefs.

  41. Cijj = 0 for i = j There are therefore N(N-1) independent centrifugal coefficients Centrifugal And Coriolis Coefficients

  42. Centrifugal And Coriolis Coefficients • Furthermore • Cijj = f(qs+1, … , qN) where s = min(i, j) • Note that Cijj characterizes the effect of the velocity • of joint j on link i • Cjjj = 0 of course • Cijj is introduced by the coordinate dependence of the • inertial coefs. dij • when i > j, • Cijj = 0 if djj is independent of qi • when i < j, • Cijj = 0 if dij is independent of qj

  43. Centrifugal And Coriolis Coefficients • Ex: Transl. joint j djj = constant, • dij = independent of qj Cijj = 0 • translational joint does not exert centrifugal forces on any link • centrifugal force is thus a unique characteristic of a rotational joint

  44. Cijm = = f(qs+1, … , qN) where s = min(i, j) Centrifugal And Coriolis Coefficients • Coriolis Coefficients: • Based again upon the functional dependence of the inertial • coefficients, the coriolis coefs Cijm for j  m are:

  45. Centrifugal And Coriolis Coefficients • The symmetry of the inertial coefs leads to the • symmetry of the Coriolis coefs: • Cijm = Cimj • Cijm characterizes the effect of the coupling of • velocities of links j and m on link i • There appears to be N2(N-1)/2 independent • Coriolis coefs • This number is reduced to N(N-1)(N-2)/3 by • incorporating the non-interacting & reflective • coupling properties of the Coriolis coefs. • Non-Interacting Cijj = 0 for all j  i • Reflective Coupling Cijm = - Cmji for all j  i, m

  46. Centrifugal And Coriolis Coefficients • Non-interacting property states that there is no Coriolis • force on a link due to the coupling of its velocity with • the velocity of an inner link. • Reflective coupling property states that the Coriolis coef • for link i, due to the coupling of velocities of links • j and m (j  i, m), is equal & opposite to the Coriolis coef • for link m due to the coupled velocities of links j & i.

  47. Coriolis forces account for the difference between • forces measured in rotating & non-rotating coord frames. • This diff. results from the observation that the energies of • a particle moving in 2 frames are diff. The rotation of • the first frame adds to a component which depends on the • coordinates of the particle & is proportional to angular • velocity of rotating frame. Centrifugal And Coriolis Coefficients

  48. Centrifugal And Coriolis Coefficients • The practical significance of this fact is that a pair of links, • the inner of which is translational, does not exert Coriolis • forces on any link. • Consequently, the Coriolis force is a unique characteristic • of the coupling of the velocity of a rotating link to the • velocity of an outer link.

  49. dot product Gi = mi• { i (g• zi-1) + i* (g • (zi-1 x pi)) } + ….. + mn• { i (g• zn-1) + i* (g • (zn-1 x pn)) }, for i,i+1,i+2,…N Gravitational Coefficients • For i = 0 (Rot. Joint), Gi = Gi(q1, … , qN) • For i = 1 (Trans. Joint), Gi = Gi(q1, … , qi-1) • There are 2 configurations of Practical Significance: • If link i translates in a direction  to gravity field, • Gi = 0 since g • zi-1 = 0 (cylindrical robot last link) • If link i rotates about an axis parallel to gravity field, • Gi = 0 since g • (zi-1 x pi) = 0 (SCARA, 1st 2 links)

  50. Hence, the gravitational coef of a joint characterizes the • work production of that joint in the gravity field. • In practice, counterbalances are introduced to reduce the • effect of gravity • A counterbalance is added to a link to alter its • mass distribution and reduce the corresponding • gravitational coef. Gravitational Coefficients

More Related