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Lecture 11. Kinematic models of contact Foundations of Statics Kinematic models Lecture 11. of contact Salisbury Kinematic models of contact Taxonomy of contacts Mobility and connectivity of grasp Foundations of Statics Foundations of statics Preview of statics. Foundations. Equivalence theorems. Matthew T. Mason Line of action. Poinsot’s theorem. Wrenches. Mechanics of Manipulation Spring 2010 Lecture 11. Today’s outline Kinematic models of contact Foundations of Statics Kinematic models of contact Kinematic models of contact Salisbury Salisbury Taxonomy of contacts Taxonomy of contacts Mobility and connectivity of grasp Mobility and connectivity of grasp Foundations of statics Preview of statics. Foundations. Foundations of statics Equivalence theorems. Preview of statics. Line of action. Poinsot’s theorem. Foundations. Wrenches. Equivalence theorems. Line of action. Poinsot’s theorem. Wrenches. Lecture 11. Kinematic models of contact Kinematic models of contact Foundations of Statics Kinematic models of contact A grasp is like a Salisbury kinematic Taxonomy of contacts Mobility and connectivity of grasp mechanism. Foundations of Assume ﬁngers statics Preview of statics. do not lift or slip. Foundations. Equivalence theorems. Line of action. Model each Poinsot’s theorem. Wrenches. contact as a spherical joint. Apply Grübler’s formula! Lecture 11. Taxonomy of contact types Kinematic models of contact Foundations of Statics In previous slide, Kinematic models No contact Point contact of contact contact was 6 freedoms without friction 5 freedoms Salisbury Taxonomy of contacts modeled as Mobility and connectivity of grasp spherical joint. Foundations of Line contact Point contact Are there other without friction with friction statics Preview of statics. 4 freedoms 3 freedoms possibilities? Foundations. Equivalence theorems. Line of action. Salisbury’s PhD Poinsot’s theorem. Planar contact Soft finger Wrenches. thesis, 1982, without friction 3 freedoms 2 freedoms included a taxonomy. Line contact Planar contact Terminology was with friction 1 freedom with friction 0 freedoms widely adopted. Lecture 11. Review of mobility and connectivity Kinematic models of contact Foundations of Statics Kinematic models of contact Salisbury Taxonomy of contacts Mobility and connectivity of grasp Foundations of statics Next several slides are repeated from Lecture 4. Preview of statics. Foundations. Equivalence theorems. Line of action. Poinsot’s theorem. Wrenches. Lecture 11. Review: Constraint and kinematic Kinematic models of contact mechanisms Foundations of Statics Kinematic models of contact Link: a rigid body; Salisbury Taxonomy of contacts Mobility and connectivity of Joint: imposes one or grasp Planar Spherical Foundations of more constraints on 3 freedoms 3 freedoms statics the relative motion of Preview of statics. Foundations. two links; Equivalence theorems. Line of action. Cylindrical Revolute Poinsot’s theorem. Kinematic 2 freedoms 1 freedom Wrenches. mechanism: a bunch of links joined by joints; Prismatic Helical 1 freedom 1 freedom lower pairs joints involving positive contact area. Lecture 11. Review: Mobility and connectivity Kinematic models of contact Foundations of Statics Kinematic models mobility of a mechanism: DOFs of contact with one link ﬁxed. Salisbury Taxonomy of contacts Mobility and connectivity of connectivity DOFs of one link grasp Foundations of relative to another. L4 statics What is the mobility of the ﬁve bar L3 Preview of statics. Foundations. Equivalence theorems. linkage at right? Line of action. L5 Poinsot’s theorem. What is the connectivity of L2 L1 Wrenches. Link 1 relative to link two? Link 3 relative to link 1? Link 3 relative to link 4? Lecture 11. Review: Mobility and connectivity Kinematic models of contact Foundations of Statics Kinematic models mobility of a mechanism: DOFs of contact with one link ﬁxed. Salisbury Taxonomy of contacts Mobility and connectivity of connectivity DOFs of one link grasp Foundations of relative to another. L4 statics What is the mobility of the ﬁve bar L3 Preview of statics. Foundations. Equivalence theorems. linkage at right? Two. Line of action. L5 Poinsot’s theorem. What is the connectivity of L2 L1 Wrenches. Link 1 relative to link two? Link 3 relative to link 1? Link 3 relative to link 4? Lecture 11. Review: Mobility and connectivity Kinematic models of contact Foundations of Statics Kinematic models mobility of a mechanism: DOFs of contact with one link ﬁxed. Salisbury Taxonomy of contacts Mobility and connectivity of connectivity DOFs of one link grasp Foundations of relative to another. L4 statics What is the mobility of the ﬁve bar L3 Preview of statics. Foundations. Equivalence theorems. linkage at right? Two. Line of action. L5 Poinsot’s theorem. What is the connectivity of L2 L1 Wrenches. Link 1 relative to link two? One. Link 3 relative to link 1? Link 3 relative to link 4? Lecture 11. Review: Mobility and connectivity Kinematic models of contact Foundations of Statics Kinematic models mobility of a mechanism: DOFs of contact with one link ﬁxed. Salisbury Taxonomy of contacts Mobility and connectivity of connectivity DOFs of one link grasp Foundations of relative to another. L4 statics What is the mobility of the ﬁve bar L3 Preview of statics. Foundations. Equivalence theorems. linkage at right? Two. Line of action. L5 Poinsot’s theorem. What is the connectivity of L2 L1 Wrenches. Link 1 relative to link two? One. Link 3 relative to link 1? Two. Link 3 relative to link 4? Lecture 11. Review: Mobility and connectivity Kinematic models of contact Foundations of Statics Kinematic models mobility of a mechanism: DOFs of contact with one link ﬁxed. Salisbury Taxonomy of contacts Mobility and connectivity of connectivity DOFs of one link grasp Foundations of relative to another. L4 statics What is the mobility of the ﬁve bar L3 Preview of statics. Foundations. Equivalence theorems. linkage at right? Two. Line of action. L5 Poinsot’s theorem. What is the connectivity of L2 L1 Wrenches. Link 1 relative to link two? One. Link 3 relative to link 1? Two. Link 3 relative to link 4? One. Lecture 11. Review: Grübler’s formula Kinematic models of contact Foundations of Given n links joined by g joints, Statics with ui constraints and fi freedoms at joint i. (Note that Kinematic models ui + fi = 6.) of contact Salisbury Assume one link is ﬁxed and constraints are all Taxonomy of contacts Mobility and connectivity of grasp independent. Foundations of statics The mobility M is Preview of statics. Foundations. Equivalence theorems. M = 6(n − 1) − ui Line of action. Poinsot’s theorem. Wrenches. = 6(n − 1) − (6 − fi ) = 6(n − g − 1) + fi Or, for a planar mechanism: M = 3(n − 1) − ui = 3(n − g − 1) + fi Lecture 11. Review: Grübler: special case for loops Kinematic models of contact Foundations of The previous formula works (sort of) for all mechanisms. Statics For loops there is a variant. Kinematic models One loop: n = g, so of contact Salisbury Taxonomy of contacts M= fi + 6(−1) Mobility and connectivity of grasp Foundations of Two loops: make a second loop by adding k links and statics Preview of statics. k + 1 joints: Foundations. Equivalence theorems. M= fi + 6(−2) Line of action. Poinsot’s theorem. Wrenches. Every loop increases excess of joints over links by 1. For l loops: M= fi − 6l for a spatial linkage, and M= fi − 3l for a planar linkage. Lecture 11. Review: Common sense Kinematic models of contact Foundations of Statics Example: what is the mobility of Watt’s linkage? Kinematic models Planar Grübler’s formula: of contact Salisbury Taxonomy of contacts Mobility and connectivity of M = 3(n − 1) − ui = 5 grasp 3 Foundations of statics M = 3(n − g − 1) + fi = 3 5 Preview of statics. 10 Foundations. M= fi − 3l = Equivalence theorems. Line of action. Poinsot’s theorem. Independent Wrenches. Spatial Grübler’s formula: constraints is a very strong M = 6(n − 1) − ui = assumption. M = 6(n − g − 1) + fi = M= fi − 6l = Why? Lecture 11. Review: Common sense Kinematic models of contact Foundations of Statics Example: what is the mobility of Watt’s linkage? Kinematic models Planar Grübler’s formula: of contact Salisbury Taxonomy of contacts Mobility and connectivity of M = 3(n − 1) − ui = 1 5 grasp 3 Foundations of statics M = 3(n − g − 1) + fi = 3 5 Preview of statics. 10 Foundations. M= fi − 3l = Equivalence theorems. Line of action. Poinsot’s theorem. Independent Wrenches. Spatial Grübler’s formula: constraints is a very strong M = 6(n − 1) − ui = assumption. M = 6(n − g − 1) + fi = M= fi − 6l = Why? Lecture 11. Review: Common sense Kinematic models of contact Foundations of Statics Example: what is the mobility of Watt’s linkage? Kinematic models Planar Grübler’s formula: of contact Salisbury Taxonomy of contacts Mobility and connectivity of M = 3(n − 1) − ui = 1 5 grasp 3 Foundations of statics M = 3(n − g − 1) + fi = 1 3 5 Preview of statics. 10 Foundations. M= fi − 3l = Equivalence theorems. Line of action. Poinsot’s theorem. Independent Wrenches. Spatial Grübler’s formula: constraints is a very strong M = 6(n − 1) − ui = assumption. M = 6(n − g − 1) + fi = M= fi − 6l = Why? Lecture 11. Review: Common sense Kinematic models of contact Foundations of Statics Example: what is the mobility of Watt’s linkage? Kinematic models Planar Grübler’s formula: of contact Salisbury Taxonomy of contacts Mobility and connectivity of M = 3(n − 1) − ui = 1 5 grasp 3 Foundations of statics M = 3(n − g − 1) + fi = 1 3 5 Preview of statics. 10 Foundations. M= fi − 3l = 1 Equivalence theorems. Line of action. Poinsot’s theorem. Independent Wrenches. Spatial Grübler’s formula: constraints is a very strong M = 6(n − 1) − ui = assumption. M = 6(n − g − 1) + fi = M= fi − 6l = Why? Lecture 11. Review: Common sense Kinematic models of contact Foundations of Statics Example: what is the mobility of Watt’s linkage? Kinematic models Planar Grübler’s formula: of contact Salisbury Taxonomy of contacts Mobility and connectivity of M = 3(n − 1) − ui = 1 5 grasp 3 Foundations of statics M = 3(n − g − 1) + fi = 1 3 5 Preview of statics. 10 Foundations. M= fi − 3l = 1 Equivalence theorems. Line of action. Poinsot’s theorem. Independent Wrenches. Spatial Grübler’s formula: constraints is a very strong M = 6(n − 1) − ui = − 2 assumption. M = 6(n − g − 1) + fi = M= fi − 6l = Why? Lecture 11. Review: Common sense Kinematic models of contact Foundations of Statics Example: what is the mobility of Watt’s linkage? Kinematic models Planar Grübler’s formula: of contact Salisbury Taxonomy of contacts Mobility and connectivity of M = 3(n − 1) − ui = 1 5 grasp 3 Foundations of statics M = 3(n − g − 1) + fi = 1 3 5 Preview of statics. 10 Foundations. M= fi − 3l = 1 Equivalence theorems. Line of action. Poinsot’s theorem. Independent Wrenches. Spatial Grübler’s formula: constraints is a very strong M = 6(n − 1) − ui = − 2 assumption. M = 6(n − g − 1) + fi = − 2 M= fi − 6l = Why? Lecture 11. Review: Common sense Kinematic models of contact Foundations of Statics Example: what is the mobility of Watt’s linkage? Kinematic models Planar Grübler’s formula: of contact Salisbury Taxonomy of contacts Mobility and connectivity of M = 3(n − 1) − ui = 1 5 grasp 3 Foundations of statics M = 3(n − g − 1) + fi = 1 3 5 Preview of statics. 10 Foundations. M= fi − 3l = 1 Equivalence theorems. Line of action. Poinsot’s theorem. Independent Wrenches. Spatial Grübler’s formula: constraints is a very strong M = 6(n − 1) − ui = − 2 assumption. M = 6(n − g − 1) + fi = − 2 M= fi − 6l = − 2 Why? Lecture 11. Applying mobility and connectivity to grasping Kinematic models of contact Foundations of Statics Salisbury suggests four measures: Kinematic models of contact M Mobility of the entire system with the ﬁnger joints Salisbury Taxonomy of contacts free. Mobility and connectivity of grasp M Mobility of the entire system, with the ﬁnger joints Foundations of statics locked. Preview of statics. Foundations. C Connectivity of the object relative to a ﬁxed palm, Equivalence theorems. Line of action. with the ﬁnger joints free. Poinsot’s theorem. Wrenches. C Connectivity of the object relative to a ﬁxed palm, with the ﬁnger joints locked. If C = 6 then object can make general motions. If C ≤ 0 then hand can immobilize object. Lecture 11. Example: the Salisbury hand Kinematic models of contact Foundations of Statics Kinematic models of contact What is C? Salisbury Taxonomy of contacts Mobility and connectivity of What is C ? grasp Foundations of statics Preview of statics. This assumes no ﬁnger is in a singular conﬁguration, Foundations. Equivalence theorems. Line of action. and contacts are not collinear. Poinsot’s theorem. Wrenches. This neglects stability of the grasp. You need statics to even start on grasp stability. Salisbury’s analysis generalizes nicely: to freely manipulate an object in the hand with point ﬁngers, the hand mechanism needs at least nine DOFs. Lecture 11. Example: the Salisbury hand Kinematic models of contact Foundations of Statics Kinematic models of contact What is C? 6 Salisbury Taxonomy of contacts Mobility and connectivity of What is C ? grasp Foundations of statics Preview of statics. This assumes no ﬁnger is in a singular conﬁguration, Foundations. Equivalence theorems. Line of action. and contacts are not collinear. Poinsot’s theorem. Wrenches. This neglects stability of the grasp. You need statics to even start on grasp stability. Salisbury’s analysis generalizes nicely: to freely manipulate an object in the hand with point ﬁngers, the hand mechanism needs at least nine DOFs. Lecture 11. Example: the Salisbury hand Kinematic models of contact Foundations of Statics Kinematic models of contact What is C? 6 Salisbury Taxonomy of contacts Mobility and connectivity of What is C ? 0 grasp Foundations of statics Preview of statics. This assumes no ﬁnger is in a singular conﬁguration, Foundations. Equivalence theorems. Line of action. and contacts are not collinear. Poinsot’s theorem. Wrenches. This neglects stability of the grasp. You need statics to even start on grasp stability. Salisbury’s analysis generalizes nicely: to freely manipulate an object in the hand with point ﬁngers, the hand mechanism needs at least nine DOFs. Lecture 11. Preview of statics Kinematic models of contact Foundations of Statics Kinematic models We will adopt Newton’s hypothesis that particles of contact interact through forces. Salisbury Taxonomy of contacts Mobility and connectivity of We can then show that rigid bodies interact through grasp Foundations of wrenches. statics Preview of statics. Screw theory applies to wrenches. Foundations. Equivalence theorems. Wrenches and twists are dual. Line of action. Poinsot’s theorem. We also get: Wrenches. Line of force; Screw coordinates applied to statics; Reciprocal product of twist and wrench; Zero Moment Point (ZMP), and its generalization. Lecture 11. What is force? Kinematic models of contact Foundations of Statics Kinematic models of contact Salisbury You cannot measure force, only its effects: Taxonomy of contacts Mobility and connectivity of deformation of structures, acceleration. grasp Foundations of We could start from Newton’s laws, but instead we statics hypothesize: Preview of statics. Foundations. A force applied to a particle is a vector. Equivalence theorems. Line of action. The motion of a particle is determined by the vector Poinsot’s theorem. Wrenches. sum of all applied forces. A particle remains at rest only if that vector sum is zero. Lecture 11. Moment of force about a line Kinematic models of contact Foundations of Statics Kinematic models of contact Salisbury Deﬁnition Taxonomy of contacts Mobility and connectivity of grasp Let l be line through origin with direction ˆ l, Foundations of statics Preview of statics. Let f act at x. Foundations. Equivalence theorems. Then the moment of force (or the torque) of f about l Line of action. Poinsot’s theorem. is given by: Wrenches. nl = ˆ · (x × f) l Lecture 11. Moment of force about a point Kinematic models of contact Foundations of Statics Deﬁnition Kinematic models Let l be line through origin with direction ˆ l, of contact Salisbury Let f act at x. Taxonomy of contacts Mobility and connectivity of grasp Then the moment of force (or the torque) of f about Foundations of O is given by: statics Preview of statics. Foundations. Equivalence theorems. nO = (x − O) × f Line of action. Poinsot’s theorem. Wrenches. If the origin is O this reduces to n = x × f. If n is moment about the origin, and nl is moment about l, and l passes through the origin, nl = ˆ · n l Lecture 11. Total force and moment Kinematic models of contact Foundations of Statics Consider a rigid body, and a system of forces {fi } acting at {xi } resp. Kinematic models of contact Salisbury Taxonomy of contacts Deﬁnition Mobility and connectivity of grasp The total force F is the sum of all external forces. Foundations of statics Preview of statics. Foundations. F= fi Equivalence theorems. Line of action. Poinsot’s theorem. Wrenches. Deﬁnition The total moment N is the sum of all corresponding moments. N= xi × fi Lecture 11. Equivalent systems of forces Kinematic models of contact Foundations of Statics Kinematic models of contact We now develop some equivalence theorems, Salisbury comparable to (or dual to) our earlier results in Taxonomy of contacts Mobility and connectivity of grasp kinematics. Foundations of statics Preview of statics. Deﬁnition Foundations. Equivalence theorems. Two systems of forces are equivalent if they have equal Line of action. Poinsot’s theorem. total force F and total moment N. Wrenches. Equivalent, speciﬁcally, because they would have the same effect on a rigid body, according to Newton. Lecture 11. Resultant Kinematic models of contact Foundations of Statics Kinematic models of contact Salisbury Taxonomy of contacts Deﬁnition Mobility and connectivity of grasp Foundations of The resultant of a system of forces is a system statics Preview of statics. comprising a single force, equivalent to the given system. Foundations. Equivalence theorems. Line of action. Poinsot’s theorem. A question: does every system of forces have a Wrenches. resultant? Lecture 11. Line of action Kinematic models of contact Foundations of Statics Consider a force f applied at x1 f some point x1 . f Kinematic models x2 of contact Total force: F = f F Salisbury Taxonomy of contacts Mobility and connectivity of Total moment: N = x1 × f. N line of action grasp Foundations of statics Consider line parallel to f through x1 , and a second Preview of statics. Foundations. point x2 on the line. Equivalence theorems. Line of action. Force f through x2 is equivalent to force f through x1 . Poinsot’s theorem. Wrenches. So point of application is more than you need to know . . . Deﬁnition The line of action of a force is a line through the point of application, parallel to the force. Lecture 11. Bound and free vectors Kinematic models of contact Foundations of Statics Kinematic models of contact Salisbury Taxonomy of contacts When you ﬁrst learned about vectors (in high Mobility and connectivity of grasp school?) you learned they aren’t attached anywhere. Foundations of We refer to those as free vectors. statics Preview of statics. Foundations. We can also deﬁne bound vectors, speciﬁcally a Equivalence theorems. Line of action. vector bound to a point, called a point vector, and a Poinsot’s theorem. Wrenches. vector bound to a line, called a line vector. So a force is a line vector. Lecture 11. Resultant of two forces Kinematic models of contact Foundations of Statics Let f1 and f2 act along L1 and Kinematic models of contact L2 respectively. Salisbury Taxonomy of contacts Slide f1 and f2 along their L1 Mobility and connectivity of grasp f1 respective lines of action to f1 + f2 Foundations of statics the intersection (if any) f2 Preview of statics. L2 Foundations. Resultant: the vector sum Equivalence theorems. Line of action. f1 + f2 , acting at the Poinsot’s theorem. Wrenches. intersection. So almost every system of forces in the plane has a resultant. Sort of like how almost every motion is a rotation. Can it be extended? Does every system of forces have a resultant? Lecture 11. Change of reference Kinematic models of contact Foundations of Statics Using reference Q or R, a system is described by Kinematic models of contact Salisbury FQ = fi NQ = (xi − Q) × fi Taxonomy of contacts Mobility and connectivity of grasp FR = fi NR = (xi − R) × fi Foundations of statics Preview of statics. From which it follows Foundations. Equivalence theorems. Line of action. Poinsot’s theorem. FR =FQ Wrenches. NR − NQ = (Q − R) × fi which gives NR =NQ + (Q − R) × F Lecture 11. Couple Kinematic models of contact Is a moment like a force? Can you apply a moment? Foundations of Statics Does it have a line of action? Kinematic models Deﬁnition of contact Salisbury A couple is a system of forces whose total force F = fi Taxonomy of contacts Mobility and connectivity of is zero. grasp Foundations of statics So a couple is a pure moment. Preview of statics. Foundations. Notice that the moment N of a couple is independent Equivalence theorems. Line of action. of reference point. N is a free vector. Poinsot’s theorem. Wrenches. Does a couple have a resultant? No! This answers the previous question: Not every system of forces has a resultant. For an arbitrary couple, can you construct an equivalent system of just two forces? Lecture 11. Equivalence theorems Kinematic models of contact Foundations of Our goal: to deﬁne a wrench, and show that every Statics system of forces is equivalent to a wrench. Kinematic models Analogous to the program for kinematics, resulting in of contact deﬁnition of twist. Salisbury Taxonomy of contacts Mobility and connectivity of grasp Theorem Foundations of statics For any reference point Q, any system of forces is Preview of statics. Foundations. equivalent to a single force through Q, plus a couple. Equivalence theorems. Line of action. Poinsot’s theorem. Wrenches. Proof. Let F be the total force; let NQ be the total moment about Q. Let new system be F at Q, plus a couple with moment NQ . Lecture 11. Two forces are sufﬁcient Kinematic models of contact Foundations of Statics Theorem Kinematic models of contact Every system of forces is equivalent to a system of just Salisbury Taxonomy of contacts two forces. Mobility and connectivity of grasp Foundations of statics Proof. Preview of statics. Foundations. Equivalence theorems. Given arbitrary F and N, construct equivalent force Line of action. Poinsot’s theorem. and couple, comprising three forces in total. Wrenches. Move couple so that one of its forces acts at same point as F. Replace those two forces with their resultant. Lecture 11. Planar system with nonzero F has a resultant Kinematic models of contact Foundations of Statics Theorem A system consisting of a single non-zero force plus a Kinematic models of contact couple in the same plane, i.e. a torque vector Salisbury perpendicular to the force, has a resultant. Taxonomy of contacts Mobility and connectivity of grasp Foundations of Proof. statics Preview of statics. Foundations. Equivalence theorems. Let F be the force, acting at P. Line of action. Poinsot’s theorem. Let N be the moment of the F Wrenches. couple. Construct an equivalent couple as in the ﬁgure. F N /F Translate the couple so −F is applied at P. Lecture 11. Poinsot’s theorem Kinematic models of contact Foundations of Theorem (Poinsot) Statics Every system of forces is equivalent to a single force, Kinematic models plus a couple with moment parallel to the force. of contact Salisbury Taxonomy of contacts Mobility and connectivity of Proof. grasp Foundations of statics Let F and N be the given force and moment. We can Preview of statics. Foundations. assume nonzero F, else the theorem is trivially true. Equivalence theorems. Line of action. Decompose the moment: N parallel to F, and N⊥ Poinsot’s theorem. Wrenches. perpendicular to F. Since planar system with nonzero force has a resultant, replace F and N⊥ by a single force F parallel to F. The desired system is F plus a couple with moment N . Lecture 11. Wrench Kinematic models of contact Foundations of Statics Kinematic models Deﬁnition of contact Salisbury Taxonomy of contacts A wrench is a screw plus a scalar magnitude, giving a Mobility and connectivity of grasp force along the screw axis plus a moment about the Foundations of screw axis. statics Preview of statics. Foundations. Equivalence theorems. The force magnitude is the wrench magnitude, and Line of action. Poinsot’s theorem. the moment is the twist magnitude times the pitch. Wrenches. Thus the pitch is the ratio of moment to force. Poinsot’s theorem is succinctly stated: every system forces is equivalent to a wrench along some screw. Lecture 11. Screw coordinates for wrenches Kinematic models of contact Foundations of Statics Let f be the magnitude of the force acting along a Kinematic models line l, of contact Salisbury Let n be the magnitude of the moment about l. Taxonomy of contacts Mobility and connectivity of grasp The magnitude of the wrench is f . Foundations of statics Recall deﬁnition in terms of Plücker coordinates: Preview of statics. Foundations. Equivalence theorems. w = fq Line of action. Poinsot’s theorem. w0 = f q0 + fpq Wrenches. where (q, q0 ) are the normalized Plücker coordinates of the wrench axis l, and p is the pitch, which is deﬁned to be p = n/f Lecture 11. Screw coordinates for wrenches demystiﬁed Kinematic models of contact Foundations of Let r be some point on the wrench axis Statics q0 = r × q Kinematic models of contact With some substitutions . . . Salisbury Taxonomy of contacts Mobility and connectivity of grasp w=f Foundations of w0 = r × f + n statics Preview of statics. Foundations. which can be written: Equivalence theorems. Line of action. Poinsot’s theorem. Wrenches. w=f w0 = n0 where n0 is just the moment of force at the origin. Screw coordinates of a wrench are actually a familiar representation (f, n0 ). Wrenches form a vector space. You can scale and add them, just as with differential twists. Lecture 11. Reciprocal product of twist and wrench Kinematic models of contact Foundations of Statics Kinematic models of contact Reciprocal product: Salisbury Taxonomy of contacts Mobility and connectivity of (ω, v0 ) ∗ (f, n0 ) = f · v0 + n0 · ω grasp Foundations of statics The power produced by the wrench (f, n0 ) and differential Preview of statics. Foundations. twist (ω, v0 ). Equivalence theorems. Line of action. Poinsot’s theorem. A differential twist is reciprocal to a wrench if and only if Wrenches. no power would be produced. Repelling if and only if positive power. Contrary if and only if negative power. Lecture 11. Force versus motion Kinematic models of contact Foundations of Statics Wrench coordinates and twist coordinates seem to Kinematic models of contact use different conventions: Salisbury Taxonomy of contacts For twists, rotation is ﬁrst. For wrenches, the Mobility and connectivity of grasp opposite. Foundations of For twists, pitch is translation over rotation, the statics Preview of statics. opposite. Foundations. Equivalence theorems. But these seeming inconsistencies are not a peculiar Line of action. Poinsot’s theorem. convention. They reﬂect deep differences between Wrenches. kinematics and statics. For example, consider the meaning of screw axis—the line—in kinematics and in statics. In kinematics, it is a rotation axis. In statics, it is a line of force. Lecture 11. Comparing motion and force Kinematic models of contact Foundations of Motion Force Statics A zero-pitch twist is a pure A zero-pitch wrench is a Kinematic models rotation. pure force. of contact Salisbury For a pure translation, the For a pure moment, the Taxonomy of contacts Mobility and connectivity of grasp direction of the axis is de- direction of the axis is de- Foundations of termined, but the location termined, but the location statics Preview of statics. is not. is not. Foundations. Equivalence theorems. Line of action. A differential translation is A couple is equivalent to Poinsot’s theorem. Wrenches. equivalent to a rotation a force along a line at about an axis at inﬁnity. inﬁnity. In the plane, any motion In the plane, any system can be described as a ro- of forces reduces to a sin- tation about some point, gle force, possibly at inﬁn- possibly at inﬁnity. ity.

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