Outline. Outline. Vector Spaces, Twists and Wrenches. Definition. ummer Screws 2009

Transcription

1 Vector Spaces, Twists and Wrenches Dimiter Zlatanov DIMEC University of Genoa Genoa, Italy Z-2 Definition A vector space (or linear space) V over the field A set V u,v,w,... of vectors with 2 operations: vector addition V V V, (u,v) u+v A1 (u+v)+w=u+(v+w) associative law A2 u+v=v+u commutative law A3 o s.t. u o+u=u zero vector A4 u u s.t. ( u)+u=o opposite vector scalar multiplication R V V, (λ,u) λu A5 λ(µu) =(λµ)u associative law A6 (λ+µ)u)=λu+µu distributive law A7 λ(u+v)=λu+λv distributive law A8 1u=u unit scalar R Z-3 Z-4

2 Trivial:,{o} Simple: numbers Q, R, C, R Q ; functions{f f : X V} Fundamental: n-tuples R n (x 1,..., x n ) v.a and s.m. : component-wise arrows from a point in space, magnitude and direction v.a. : parallelogram rule a a+ b s.m.: length dilation b a λ a Forces acting on a particle, F 3 v.a : resultant force (parallelogram rule) s.m: proportional change of force intensity arrow from the particle, magnitude and direction Force fields f(p) mappings { f f : E 3 F 3 } arrow at every point P Velocities of a (free) particle M 3 Velocity fields v(p) Velocities and forces? f+ v=? v.a. must be defined for every pair of vectors Z-5 Z-6 Car traffic through an intersection arrow with magnitude= average # of cars in a direction s.m. : clear; v.a. : parallelogram rule? 40 cars/h 30 cars/h 50 cars/h? Reaction forces acting on a rigid body with a fixed point v.a. : resultant force (parallelogram rule) s.m.: proportional change of force intensity Forces acting on a rigid body v.a.: resultant force? two forces have a resultant force only if their axes intersect a vector space must be closed under v.a. Not everything with magnitude and direction is a vector Z-7 Z-8

3 Couples of forces acting on a rigid body v.a. : resultant couple (parallelogram rule) s.m.: proportional change of the moment magnitude Forces and couples acting on a rigid body v.a.: resultant force or couple? two forces have a resultant force or couple only if their axes are coplanar a vector space must be closed under v.a. Instantaneous rotations of a body with a fixed point v.a. : resultant rotation, (parallelogram rule) obtained when the body is the end-effector of an RR chain with intersecting axes s.m.: proportional change of rotation amplitude Instantaneous rotations of a rigid body v.a.: resultant rotation? two rotation have a resultant rotation only if their axes intersect a vector space must be closed under v.a. Z-9 Z-10 Instantaneous translations of a rigid body v.a. : resultant translation, (parallelogram rule) as if the body is the end-effector of a PP chain s.m.: proportional change of translation speed Instantaneous rotations and translations of a rigid body v.a.: resultant rotation? two rotation have a resultant rotation or translation only if their axes are coplanar a vector space must be closed under v.a. Z-11 Z-12

4 Systems of Forces External action on a system of particles B: { f P P B} System of forces at O acting on a rigid body: Φ O ={ϕ O,µ O } force ϕ O thru O with intensity and direction f couple µ O with moment m O applied in parallel to the body described by ( f, m O ) at O Instantaneous Motions Instantaneous motion of a particle system B: { v P P B} Instantaneous Motion at O : Υ O ={ O,τ O } inst. rotation thru O O with amplitude and direction ω inst. translation τ O with velocity v O applied in series to the body described by ( ω, v O ) at O Fundamental fact of statics : all external actions on a rigid body are of this type [this is an axiom in rigid body dynamics] Fundamental fact of rigid body velocity kinematics: all instantaneous motions are of this type [this is a theorem in position kinematics] Z-13 Z-14 Wrench Φ O ={ϕ O,µ O }, ( f, m O ) and Φ O ={ϕ O,µ O }, ( f, m O ) are equivalent iff f = f and m O = m O + OO f (the shifting law) The equivalence class, ζ=[φ], is called a wrench Twist Υ O ={ O,τ O }, ( ω, v O ) and are equivalent iff ω= ω and v O = v O + OO ω (the shifting law) Υ O ={ O,τ O }, ( ω, v O ) The equivalence class, ξ=[υ], is called a twist A wrench is a system of forces (reduced at a point) with equivalent systems identified A wrench is an entity invariant of frame choice For a given origin, O, it is given by a pair of vectors: ζ=( f, m O ) the resultant force and moment at O A twist is an instantaneous motion (reduced at a point) with equivalent motions identified A twist is an entity invariant of frame choice For a given origin, O, it is given by a pair of vectors: ξ=( ω, v O ) the body angular velocity and the velocity of the point coinciding with O Z-15 Z-16

5 Wrench and Twist Spaces Wrenches form a vector space, F 6, se*(3): v.a. : resultant wrench: all forces acting in parallel at O, ζ+ζ =( f, m O )+( f, m O)=( f+ f, m O + m O) s.m. : proportional increase of intensity at O, λζ= λ( f, m O )=(λf, λ m O ) [show that + and. do not depend on O] Twists form a vector space, M 6, se(3): v.a. : resultant motion: all motions acting in series (in any order!) at O, ξ+ξ =( ω, v O )+( ω, v O)=( ω+ ω, v O + v O) s.m. : proportional increase of amplitude at O, λξ=λ( ω, v O )=(λ ω, λ v O ) [show that + and. do not depend on O] Canonical Representative of a Twist/Wrench For any wrench, ζ=( f, m O ) if f = o then ζ=( o, m) O else! line l(ζ)={ r+ λf/ f } the screw axis s.t. P (force and moment are parallel) l(ζ) m P = h f The pitch h and the axis point closest to the origin are: h= f m O f f, Thus at O, r= f m O f f Similarly for a twist ζ=( f, h f+ r f) Z-17 Z-18 Screw A line l with a pitch h (a metric quantity) is a geometric element called a screw The screw of a couple/translation has no axis, only a direction: infinite pitch screw, h= A (geometric) screw is not a vector (why?). Screws form the projective space underlying the space of twists and wrenches [The projective space of V is obtained by identifying v λ v ] Z-19 Z-20

6 Linear Combinations Definition. The span of v 1,...,v n Vis the set of their linear combinations: Span(v 1,...,v n )={λ 1 v λ n v n λ i R} Example. The plane containing v 1, v 2 in 3D space Interpretation. Span{twists}: all end-effector motions of a serial chain Span{wrenches} : all end-effector constraints of a parallel chain Exercise. Consider a spherical RRR chain (concurrent axes). Find all possible instantaneous motions of (1) the end-effector (2) the second link Exercise. Consider a planar RR chain. Find all instantaneous motions of the end-effector. Linear bspaces Definition. W is a linear subspace of the vector space V if (1) W is a subset of V ; (2) W is a vector space with the v.a. and s.m. of V. Example. A plane thru the origin. A plane not thru the origin? Example. Planar motions se(2) se(3). Theorem. The intersection of two subspaces is a subspace. The nontrivial union of two subspaces is not a subspace. The difference of two subspaces is never a subspace (why?). Example. Impossible motions (e.g. of a planar-chain end-effector). Theorem. Span(v 1,...,v n ) is the smallest subspace with v 1,...,v n Z-21 Z-22 Linear Dependence and Independence Definition. The set {v 1,...,v n } is linearly dependent if (λ 1,..., λ n )=(0,...,0) s.t. λ 1 v λ n v n =0 Else it is linearly independent. Examples. When is {v 1 } l.d.? {v 1,v 2 }? When are three arrows l.d.? When are two twists l.d.? Facts. (1) If {v 1,...,v n } o then {v 1,...,v n } are l.d. (2) subset of a l.i. set is l.i. (3) superset of a l.d. set is l.d. Exercise. When are three forces l.d.? Z-23 Z-24

7 Linear Dependence and Independence Definition. The set {v 1,...,v n } is linearly dependent if (λ 1,..., λ n )=(0,...,0) s.t. λ 1 v λ n v n =0 Else it is linearly independent. Examples. When is {v 1 } l.d.? {v 1,v 2 }? When are three arrows l.d.? When are two twists l.d.? A set of forces in equilibrium? Dimension and Basis Definitions. (1) dim V < if V =Span(v 1,...,v n ) (2) dim V = n if l.i. {v 1,...,v n } (a basis) s.t. V =Span(v 1,...,v n ) Proposition. dim V = n n-basis basis is an n-basis if {v 1,...,v n } l.i. then {u,v 1,...,v n } is l.d. u 1 n 1 n Facts. (1) If {v 1,...,v n } o then {v 1,...,v n } are l.d. (2) subset of a l.i. set is l.i. (3) superset of a l.d. set is l.d. Exercises. Find a basis and establish the dimension of the rotations with axes through a point the translations the planar motions Exercise. When are three forces l.d.? Z-25 Z-26 Dimension and Basis Example. Plücker bases for a frame Oxyz The 3 unit rotations about the axes the 3 unit translations directed as the axes {ρ Ox,ρ Oy,ρ Oz,τ x,τ y,τ z } ξ=( ω, v O )=(ω x i+ωy j+ ωz k, vox i+voy j+ voz k) = ω x ρ Ox + ω y ρ Oy + ω z ρ Oz + v Ox τ x + v Oy τ y + v Oz τ z } The 3 unit forces along the axes the 3 unit couples directed as the axes {ϕ Ox,ϕ Oy,ϕ Oz,µ x,µ y,µ z } ζ=( f, m O )=(f x i+fy j+ fz k, mox i+moy j+ moz k) = f x ϕ Ox + f y ϕ Oy + f z ϕ Oz + m x µ x + m y µ y + m z µ z } [beware of unit problems! Are f x, m x dimensioned?] Dimension and Basis Example. Ball bases for a frame Oxyz 6 unit, h= ±1, twists/wrenches along the axes {ξ + Ox,ξ+ Oy,ξ+ Oz,ξ Ox,ξ Oy,ξ Oz } Example. Joint twists of a 6dof serial manipulator, 1P, 5R joints 6 unit joint translations/rotations in a nonsingular configuration {ρ 1,ρ 2,τ 3,ρ 4,ρ 5,ρ 6 } ξ= λ 1 ρ 1 + λ 2 ρ 2 + λ 3 τ 3 + λ 4 ρ 4 + λ 5 ρ 5 + λ 6 ρ 6 [ ] [ e ξ= ω 1 1 +ω r 1 e 2 1 ξ=j θ ] [ ] [ e 2 0 +v r 2 e 3 +ω 2 e 4 3 θ=(ω1, ω 2, v 3,..., ω 6 ) T are the joint speeds, with units The columns of the Jacobian are not twists They are not vectors in the same vector space Treat the Jacobian matrix with care! ] [ e 4 +ω r 4 e 5 4 λ i are dimensionless ] [ e 5 +ω r 5 e 6 5 ] e 6 r 6 e 6 Z-27 Z-28

8 Definition. Let U, W V (subspaces) U+ W ={v=u+w u U,w W} Facts. ms and Direct ms of bspaces U+ W =Span(U W) dim(u+ W)=dim U+dim W dim(u W) Definition. V = U W if (1) V = U+ W (2) U W =o Examples. Two planes in 3D A line and a plane in 3D Spherical motions and translations Spherical and planar motions Z-29 Z-30 Dual Spaces and Scalar Products Definition. The dual V * of vector space V ( dim V = n ) V ={f : V R f : linear} ( dim V = n) Fact. basis {e of V! dual basis {e 1,...,e 1,...,e n } n} defined by e i (e j)=δ ij Example. The space of forces acting on a particle is dual to the space of particle velocities. f( v)= f v Example. The wrenches, se*(3), are dual to the twists, se(3) ζ(ξ)=ζ ξ= f v O + m O ω Interpretation. The application of a wrench on a twist (also called their reciprocal product) measures the power exerted by the system of forces for the instantaneous motion Fact. Dual Spaces and Scalar Products For the Plücker twist basis The dual wrench basis is Notation. When dual bases are used, ζ ξ=ζ T ξ (interpreting as column coordinate vectors) Hence, the notation ζ ξ is also used {ρ Ox,ρ Oy,ρ Oz,τ x,τ y,τ z } {µ x,µ y,µ z,ϕ Ox,ϕ Oy,ϕ Oz } Z-31 Z-32

9 Definition. When a dual vector maps a vector into zero, the two are said to be orthogonal. Example. Exercise. Dual Spaces and Scalar Products When a wrench exerts no power on a twist, they are orthogonal (also called reciprocal). What are the geometric conditions for the following to be reciprocal: a translation and a couple a translation and a wrench/a couple and a twist a rotation and a force a twist and a wrench Dual Spaces and Scalar Products Definition. Let U be a subspace of V Then U ={w V w(u)=0, u U} is the orthogonal annihilator of U Fact. U is a subspace. (why?) Example. In Euclidian spaces V and V * are identified and we have U U = V Example. For a twist subspace U, U is a wrench subspace composed of all wrenches that exert no power on any motion in U Facts. If U se(3) then dim U + dim U = 6 Interpretation. U is a constraint system allowing only motions in U Z-33 Z-34 Dual Spaces and Scalar Products Exercises. Describe U when U is the planar motions the system spanned by the rotations in a plane the system spanned by the rotations shown Z-35 Z-36

10 Screw Systems Definition. The projective space underlying a twist or wrench subspace is called a screw system. An n-system underlies an n-dimensional subspace. Definition. Two screw systems are reciprocal when any wrench acting on a screw in one system exerts no power on any twist on a screw in the other system. Remark. A screw system can be self-reciprocal. Screw Systems The Gibson-Hunt Classification. Screw systems are labelled by: 2 or 3 Dimension I or II Contains or not screws of more than one finite pitch A,..., D The number from 0 to 3 of the independent -screws in the system Definition. Two systems are classified as identical when there is a rigid body displacement that can make them coincide. Remark. For classification purposes it is sufficient to consider only systems underlying subspaces of dimension 2 or 3. (If higher, study the reciprocal.) angle, pitch Additional parameters where needed Z-37 Z-38 Example. Screw Systems 2-IA(h x, h y ) the general two system Example. Phillips 1984 Screw Systems 3-IA(h x, h y, h z ) the general three-system Phillips 1984 The screw axes form a self-intersecting surface, the cylindroid Screw axes pass through every point in space. The same-pitch quadrics are hyperboloids of one sheet. Z-39 Z-40