5. Nonholonomic constraint Mechanics of Manipulation

Transcription

1 5. Nonholonomic constraint Mechanics of Manipulation Matt Mason Carnegie Mellon Lecture 5. Mechanics of Manipulation p.1

2 Lecture 5. Nonholonomic constraint. Chapter 1 Manipulation Case 1: Manipulation b a human Case 2: An automated assembl sstem Issues in manipulation A taonom of manipulation techniques Bibliographic notes 8 Eercises 8 Chapter 2 Kinematics Preliminaries Planar kinematics Spherical kinematics Spatial kinematics Kinematic constraint Kinematic mechanisms Bibliographic notes 36 Eercises 37 Chapter 3 Kinematic Representation Representation of spatial rotations Representation of spatial displacements Kinematic constraints Bibliographic notes 72 Eercises 72 Chapter 4 Kinematic Manipulation Path planning Path planning for nonholonomic sstems Kinematic models of contact Bibliographic notes 88 Eercises 88 Chapter 5 Rigid Bod Statics Forces acting on rigid bodies Polhedral conve cones Contact wrenches and wrench cones Cones in velocit twist space The oriented plane Instantaneous centers and Reuleau s method Line of force; moment labeling Force dual Summar Bibliographic notes 117 Eercises 118 Chapter 6 Friction Coulomb s Law Single degree-of-freedom problems Planar single contact problems Graphical representation of friction cones Static equilibrium problems Planar sliding Bibliographic notes 139 Eercises 139 Chapter 7 Quasistatic Manipulation Grasping and fituring Pushing Stable pushing Parts orienting Assembl Bibliographic notes 173 Eercises 175 Chapter 8 Dnamics Newton s laws A particle in three dimensions Moment of force; moment of momentum Dnamics of a sstem of particles Rigid bod dnamics The angular inertia matri Motion of a freel rotating bod Planar single contact problems Graphical methods for the plane Planar multiple-contact problems Bibliographic notes 207 Eercises 208 Chapter 9 Impact A particle Rigid bod impact Bibliographic notes 223 Eercises 223 Chapter 10 Dnamic Manipulation Quasidnamic manipulation Brie dnamic manipulation Continuousl dnamic manipulation Bibliographic notes 232 Eercises 235 Appendi A Infinit 237 Lecture 5. Mechanics of Manipulation p.2

3 Outline. An eample: the uniccle. Integrable and nonintegrable constraints Vector fields and distributions Frobenius s theorem Lecture 5. Mechanics of Manipulation p.3

4 Holonomic does not mean unconstrained!!! Holonomic means the constraints can be written as equations independent of q f(q, t) = 0 A mobile robot with no constraints is holonomic. A mobile robot capable of arbitrar planar velocities is holonomic. A mobile robot capable of onl translations is holonomic. Lecture 5. Mechanics of Manipulation p.4

5 Uniccle constraint The uniccle cannot move sidewas. Let ẋ q = ẏ θ and let w 1 = (sinθ, cosθ, 0) (,) θ ( sin θ cos θ ) so the constraint is written w 1 q = 0. Lecture 5. Mechanics of Manipulation p.5

6 Uniccle freedom The uniccle can move in two directions, epressed b defining g 1 (q) = 0 0 1, g 2 (q) = cosθ sinθ 0 (,) θ ( sin θ cos θ ) and noting that the uniccle s motion is (missing from book) q = u 1 g 1 + u 2 g 2 where u 1 and u 2 are arbitrar reals. The are the controls. So, how man DOFs does the uniccle have? Lecture 5. Mechanics of Manipulation p.6

7 Uniccle freedom The uniccle can move in two directions, epressed b defining g 1 (q) = 0 0 1, g 2 (q) = cosθ sinθ 0 (,) θ ( sin θ cos θ ) and noting that the uniccle s motion is (missing from book) q = u 1 g 1 + u 2 g 2 where u 1 and u 2 are arbitrar reals. The are the controls. So, how man DOFs does the uniccle have? THREE!!! Lecture 5. Mechanics of Manipulation p.6

8 Unsteered cart constraint and freedom The unsteered cart cannot turn, and cannot move sidewas. Let.;/ w 1 = (sinθ, cosθ, 0),w 2 = (0, 0, 1) so the two constraints are written w 1 q = 0, w 2 q = 0. Epanding the products: ẋsin θ ẏ cosθ = 0 θ = 0 Leaves of the foliation These can be integrated: θ = θ 0 ( 0 ) sinθ 0 ( 0 ) cosθ 0 = 0 Lecture 5. Mechanics of Manipulation p.7

9 Uniccle versus cart Uniccle. One velocit constraint. Three freedoms. Unsteered cart Two velocit constraints. Integrable. Equivalent to two configuration constraints. One freedom. (,) θ ( sin θ cos θ ) Sstem is nonholonomic if the constraint cannot be written in the form f(q, t) = 0..;/ Lecture 5. Mechanics of Manipulation p.8

10 How can ou tell? How can ou tell whether a velocit constraint is integrable? 1. Tr to integrate it for a while. 2. Determine whether the DOFs were reduced. 3. Lie brackets!!! (Frobenius s theorem) (,) θ ( sin θ cos θ ).;/ Lecture 5. Mechanics of Manipulation p.9

11 Pfaffian constraints A set of k Pfaffian constraints are of the form w i (q) q = 0, i = 1...k where the w i are linearl independent row vectors, and q is a column vector. All the velocit constraints we have considered for the uniccle and the cart are Pfaffian. Lecture 5. Mechanics of Manipulation p.10

12 Vector fields A vector field is a smooth map f(q) : C T q C from configurations q to velocit vectors q. g 1 : turning Note: In differential geometr vector sometimes means specificall velocit vector. g 2 : forward rolling Lecture 5. Mechanics of Manipulation p.11

13 Distributions A distribution is a smooth map assigning a linear subspace of T q C to each configuration q of C. Eample: The linear span of g 1 and g 2. Recall that for the uniccle _q = u 1 g 1 + u 2 g 2 for u 1, u 2 R. So the figure shows the feasible velocities for ever q. (Well, it onl shows a circular patch where it should show a whole plane at ever q.) Lecture 5. Mechanics of Manipulation p.12

14 Regular distributions and Lie brackets A distribution is regular if its dimension is constant over the configuration space. Let f, g be two vector fields on C. Define the Lie bracket [f, g] to be the vector field g q f f q g What is this thing written g q velocit w.r.t. configuration variable. f or q? Matri. Each column is partial of Lecture 5. Mechanics of Manipulation p.13

15 Lie brackets, eample. Let s take the Lie bracket [g 1, g 2 ]. g 1 q = g 2 q = sinθ 0 0 cos θ For the new vector field defined b the Lie bracket we obtain g 3 = [g 1, g 2 ] = g 2 q g 1 g 1 q g 2 sinθ = cosθ Lecture 5. Mechanics of Manipulation p.14

16 Lie brackets eample continued g 3 = sinθ cosθ Phsicall, g 3 moves sidewas. It is linearl independent of g 1 and g 2, and it violates the constraint w 1. What is its phsical significance? Given two vector fields f and g, 1. Follow f for some time ǫ; 2. Follow g for ǫ; 3. Follow f for ǫ; 4. Follow g for ǫ. In the limit as ǫ approaches zero, the result of the above motion approaches the Lie bracket [f, g]. The Lie bracket could have been called parallel parking product. Lecture 5. Mechanics of Manipulation p.15 0

17 Involutive distribution A distribution is involutive if it is closed under Lie bracket operations. The involutive closure of a distribution is the closure of the distribution under Lie bracketing. Lecture 5. Mechanics of Manipulation p.16

18 Frobenius s theorem Theorem 2.8 (Frobenius s theorem): A regular distribution is integrable if and onl if it is involutive. Proof: To prove that an integrable distribution is involutive, take the Talor series epansion of the parallel parking maneuver as a function of ǫ. The second order terms are Lie brackets! If the distribution is involutive, the Lie brackets must also be contained in the distribution. To prove that involutive distributions are integrable... nonholonomic parallel parking helps Lecture 5. Mechanics of Manipulation p.17

19 Net: Ais angle. Rodrigues s theorem. Rotation Chapter 1 Manipulation Case 1: Manipulation b a human Case 2: An automated assembl sstem Issues in manipulation A taonom of manipulation techniques Bibliographic notes 8 Eercises 8 Chapter 2 Kinematics Preliminaries Planar kinematics Spherical kinematics Spatial kinematics Kinematic constraint Kinematic mechanisms Bibliographic notes 36 Eercises 37 Chapter 3 Kinematic Representation Representation of spatial rotations Representation of spatial displacements Kinematic constraints Bibliographic notes 72 Eercises 72 Chapter 4 Kinematic Manipulation Path planning Path planning for nonholonomic sstems Kinematic models of contact Bibliographic notes 88 Eercises 88 Chapter 5 Rigid Bod Statics Forces acting on rigid bodies Polhedral conve cones Contact wrenches and wrench cones Cones in velocit twist space The oriented plane Instantaneous centers and Reuleau s method Line of force; moment labeling Force dual Summar Bibliographic notes 117 Eercises 118 Chapter 6 Friction Coulomb s Law Single degree-of-freedom problems Planar single contact problems Graphical representation of friction cones Static equilibrium problems Planar sliding Bibliographic notes 139 Eercises 139 Chapter 7 Quasistatic Manipulation Grasping and fituring Pushing Stable pushing Parts orienting Assembl Bibliographic notes 173 Eercises 175 Chapter 8 Dnamics Newton s laws A particle in three dimensions Moment of force; moment of momentum Dnamics of a sstem of particles Rigid bod dnamics The angular inertia matri Motion of a freel rotating bod Planar single contact problems Graphical methods for the plane Planar multiple-contact problems Bibliographic notes 207 Eercises 208 Chapter 9 Impact A particle Rigid bod impact Bibliographic notes 223 Eercises 223 Chapter 10 Dnamic Manipulation Quasidnamic manipulation Brie dnamic manipulation Continuousl dnamic manipulation Bibliographic notes 232 Eercises 235 Appendi A Infinit 237 Lecture 5. Mechanics of Manipulation p.18