Lecture 5 Spatial Kinematics. Constraint

Lecture 5 Spatial Kinematics. Constraint.

Spherical kinematics Euler’s theorem

Spatial kinematics Lecture 5 Classifying displacements Chasles’s theorem Screws Spatial Kinematics. Constraint. Cones

Kinematic constraint Counting variables and equations Matthew T. Mason General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula Mechanics of Manipulation Lecture 5 Outline Spatial Kinematics. Constraint.

Spherical kinematics Spherical Euler’s theorem kinematics Euler’s theorem

Spatial kinematics Spatial kinematics Classifying displacements Chasles’s theorem Classifying displacements Screws Chasles’s theorem Cones Kinematic Screws constraint Counting variables and Cones equations General form Classifying constraints Mobility, Grübler Kinematic constraint Mobility and connectivity Counting variables and equations Grübler’s formula General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula Lecture 5 About spherical kinematics Spatial Kinematics. Constraint.

Spherical kinematics Euler’s theorem

Spatial kinematics Classifying displacements Chasles’s theorem Screws I Why study motions of the sphere? Because it Cones 3 Kinematic corresponds to rotations about a given point of E . constraint Counting variables and I There is a close connection to planar kinematics. Let equations General form the radius of the sphere approach inﬁnity ... Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula Lecture 5 Two not-antipodal points enough Spatial Kinematics. Constraint.

Spherical kinematics Euler’s theorem

Spatial kinematics Classifying displacements Chasles’s theorem Theorem (2.5) Screws Cones

A displacement of the sphere is completely determined Kinematic constraint by the motion of any two points that are not antipodal. Counting variables and equations General form Classifying constraints Proof. Mobility, Grübler Mobility and connectivity Construct a coordinate frame ... Grübler’s formula Lecture 5 Euler’s theorem Spatial Kinematics. Constraint. Theorem (2.6) Spherical kinematics For every spatial rotation, there is a line of ﬁxed points. In Euler’s theorem other words, every rotation about a point is a rotation Spatial kinematics Classifying displacements about a line, called the rotation axis. Chasles’s theorem Screws Cones

Kinematic Proof. constraint Counting variables and equations 0 0 0 I Deﬁne A, ⊥AA , B, B , ⊥BB . General form Classifying constraints Mobility, Grübler I Deﬁne C to be either intersection of Mobility and connectivity ⊥AA0 with ⊥BB0. A A Grübler’s formula

I Let R be the rotation mapping A to B B 0 O A and C to itself. C 0 I Show R maps B to B , so R is the given displacement. Lecture 5 Spatial kinematics Spatial Kinematics. Constraint. Why? What do we want to know?

Spherical kinematics Euler’s theorem

Spatial kinematics Classifying displacements Chasles’s theorem Screws Cones

I Why? Kinematic constraint I We seem to live in a three-dimensional space. Counting variables and equations I What do we want to know? Let’s review the plane General form Classifying constraints and the sphere for ideas! Mobility, Grübler Mobility and connectivity Grübler’s formula Lecture 5 Spatial kinematics Spatial Kinematics. Constraint. Why? What do we want to know?

Spherical kinematics Euler’s theorem

Spatial kinematics Classifying displacements Chasles’s theorem Screws Cones

One—the identity. The null displacement. Nope.

Lecture 5 Review of displacements: planar Spatial Kinematics. Constraint. For the Euclidean plane, are there ...

I ... rotations that are not translations? Spherical kinematics I ... translations that are not rotations? Euler’s theorem I ... displacements that are both rotations and Spatial kinematics Classifying displacements translations? Chasles’s theorem Screws Cones I ... displacements that are neither? Kinematic constraint Counting variables and equations General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula

Rotations Translations

Displacements

SE(2): Displacements of the Euclidean plane Many.

One—the identity. The null displacement. Nope.

Lecture 5 Review of displacements: planar Spatial Kinematics. Constraint. For the Euclidean plane, are there ...

I ... rotations that are not translations? Many. Spherical kinematics I ... translations that are not rotations? Euler’s theorem I ... displacements that are both rotations and Spatial kinematics Classifying displacements translations? Chasles’s theorem Screws Cones I ... displacements that are neither? Kinematic constraint Counting variables and equations General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula

Rotations Translations

Displacements

SE(2): Displacements of the Euclidean plane One—the identity. The null displacement. Nope.

Lecture 5 Review of displacements: planar Spatial Kinematics. Constraint. For the Euclidean plane, are there ...

I ... rotations that are not translations? Many. Spherical kinematics I ... translations that are not rotations? Many. Euler’s theorem I ... displacements that are both rotations and Spatial kinematics Classifying displacements translations? Chasles’s theorem Screws Cones I ... displacements that are neither? Kinematic constraint Counting variables and equations General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula

Rotations Translations

Displacements

SE(2): Displacements of the Euclidean plane Nope.

Lecture 5 Review of displacements: planar Spatial Kinematics. Constraint. For the Euclidean plane, are there ...

I ... rotations that are not translations? Many. Spherical kinematics I ... translations that are not rotations? Many. Euler’s theorem I ... displacements that are both rotations and Spatial kinematics Classifying displacements translations? One—the identity. The null Chasles’s theorem Screws displacement. Cones I ... displacements that are neither? Kinematic constraint Counting variables and equations General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula

Rotations Translations

Displacements

SE(2): Displacements of the Euclidean plane Lecture 5 Review of displacements: planar Spatial Kinematics. Constraint. For the Euclidean plane, are there ...

I ... rotations that are not translations? Many. Spherical kinematics I ... translations that are not rotations? Many. Euler’s theorem I ... displacements that are both rotations and Spatial kinematics Classifying displacements translations? One—the identity. The null Chasles’s theorem Screws displacement. Cones I ... displacements that are neither? Nope. Kinematic constraint Counting variables and equations General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula

Rotations Translations

Displacements

SE(2): Displacements of the Euclidean plane Many. No.

One. Nope.

Lecture 5 Review of displacements: spherical Spatial Kinematics. Constraint. For the sphere, are there ...

I ... rotations that are not translations? Spherical kinematics I ... translations that are not rotations? Euler’s theorem Spatial kinematics I ... displacements that are both rotations and Classifying displacements Chasles’s theorem translations? Screws Cones I ... displacements that are neither? Kinematic constraint Counting variables and equations General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula

Rotations Translations

Displacements

SO(3): Displacements of the sphere No.

One. Nope.

Lecture 5 Review of displacements: spherical Spatial Kinematics. Constraint. For the sphere, are there ...

I ... rotations that are not translations? Many. Spherical kinematics I ... translations that are not rotations? Euler’s theorem Spatial kinematics I ... displacements that are both rotations and Classifying displacements Chasles’s theorem translations? Screws Cones I ... displacements that are neither? Kinematic constraint Counting variables and equations General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula

Rotations Translations

Displacements

SO(3): Displacements of the sphere One. Nope.

Lecture 5 Review of displacements: spherical Spatial Kinematics. Constraint. For the sphere, are there ...

I ... rotations that are not translations? Many. Spherical kinematics I ... translations that are not rotations? No. Euler’s theorem Spatial kinematics I ... displacements that are both rotations and Classifying displacements Chasles’s theorem translations? Screws Cones I ... displacements that are neither? Kinematic constraint Counting variables and equations General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula

Rotations Translations

Displacements

SO(3): Displacements of the sphere Nope.

Lecture 5 Review of displacements: spherical Spatial Kinematics. Constraint. For the sphere, are there ...

I ... rotations that are not translations? Many. Spherical kinematics I ... translations that are not rotations? No. Euler’s theorem Spatial kinematics I ... displacements that are both rotations and Classifying displacements Chasles’s theorem translations? One. Screws Cones I ... displacements that are neither? Kinematic constraint Counting variables and equations General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula

Rotations Translations

Displacements

SO(3): Displacements of the sphere Lecture 5 Review of displacements: spherical Spatial Kinematics. Constraint. For the sphere, are there ...

I ... rotations that are not translations? Many. Spherical kinematics I ... translations that are not rotations? No. Euler’s theorem Spatial kinematics I ... displacements that are both rotations and Classifying displacements Chasles’s theorem translations? One. Screws Cones I ... displacements that are neither? Nope. Kinematic constraint Counting variables and equations General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula

Rotations Translations

Displacements

SO(3): Displacements of the sphere Many. Many.

One. It’s not obvious!

Lecture 5 Preview of spatial displacements Spatial Kinematics. Constraint. For Euclidean three space, are there ...

Rotations Translations

Displacements

SE(3): Displacements of Euclidean three space Many.

One. It’s not obvious!

Lecture 5 Preview of spatial displacements Spatial Kinematics. Constraint. For Euclidean three space, are there ...

Rotations Translations

Displacements

SE(3): Displacements of Euclidean three space One. It’s not obvious!

Lecture 5 Preview of spatial displacements Spatial Kinematics. Constraint. For Euclidean three space, are there ...

I ... rotations that are not translations? Many. Spherical kinematics I ... translations that are not rotations? Many. Euler’s theorem Spatial kinematics I ... displacements that are both rotations and Classifying displacements Chasles’s theorem translations? Screws Cones I ... displacements that are neither? Kinematic constraint Counting variables and equations General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula

Rotations Translations

Displacements

SE(3): Displacements of Euclidean three space It’s not obvious!

Lecture 5 Preview of spatial displacements Spatial Kinematics. Constraint. For Euclidean three space, are there ...

I ... rotations that are not translations? Many. Spherical kinematics I ... translations that are not rotations? Many. Euler’s theorem Spatial kinematics I ... displacements that are both rotations and Classifying displacements Chasles’s theorem translations? One. Screws Cones I ... displacements that are neither? Kinematic constraint Counting variables and equations General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula

Rotations Translations

Displacements

SE(3): Displacements of Euclidean three space Lecture 5 Preview of spatial displacements Spatial Kinematics. Constraint. For Euclidean three space, are there ...

I ... rotations that are not translations? Many. Spherical kinematics I ... translations that are not rotations? Many. Euler’s theorem Spatial kinematics I ... displacements that are both rotations and Classifying displacements Chasles’s theorem translations? One. Screws Cones I ... displacements that are neither? It’s not obvious! Kinematic constraint Counting variables and equations General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula

Rotations Translations

Displacements

SE(3): Displacements of Euclidean three space No. It’s not a rotation.

No. It’s not a translation.

Lecture 5 What is a twist? Spatial Kinematics. Constraint.

Spherical kinematics Euler’s theorem Deﬁnition (twist) Spatial kinematics Classifying displacements Chasles’s theorem A twist is a rotation about a line, Screws composed with a translation parallel to Cones Kinematic the line. constraint Counting variables and equations General form I Does a twist have a ﬁxed point? Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula I Does a twist move all points along parallel lines? No. It’s not a translation.

Lecture 5 What is a twist? Spatial Kinematics. Constraint.

Spherical kinematics Euler’s theorem Deﬁnition (twist) Spatial kinematics Classifying displacements Chasles’s theorem A twist is a rotation about a line, Screws composed with a translation parallel to Cones Kinematic the line. constraint Counting variables and equations General form I Does a twist have a ﬁxed point? Classifying constraints Mobility, Grübler No. It’s not a rotation. Mobility and connectivity Grübler’s formula I Does a twist move all points along parallel lines? Lecture 5 What is a twist? Spatial Kinematics. Constraint.

Spherical kinematics Euler’s theorem Deﬁnition (twist) Spatial kinematics Classifying displacements Chasles’s theorem A twist is a rotation about a line, Screws composed with a translation parallel to Cones Kinematic the line. constraint Counting variables and equations General form I Does a twist have a ﬁxed point? Classifying constraints Mobility, Grübler No. It’s not a rotation. Mobility and connectivity Grübler’s formula I Does a twist move all points along parallel lines? No. It’s not a translation. Twists! No. Chasles’s theorem.

Lecture 5 Preview of spatial displacements Spatial Kinematics. Constraint. For Euclidean three space, are there ...... rotations that are not translations? Many. I Spherical I ... translations that are not rotations? Many. kinematics Euler’s theorem

I ... displacements that are both rotations and Spatial kinematics translations? One. Classifying displacements Chasles’s theorem I ... displacements that are neither? Screws Cones

I ... displacements that are not twists? Kinematic constraint Counting variables and equations General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula

Rotations Translations

Displacements

SE(3): Displacements of Euclidean three space No. Chasles’s theorem.

I ... displacements that are both rotations and Spatial kinematics translations? One. Classifying displacements Chasles’s theorem I ... displacements that are neither? Twists! Screws Cones

I ... displacements that are not twists? Kinematic constraint Counting variables and equations General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula

Rotations Translations

Displacements

SE(3): Displacements of Euclidean three space Lecture 5 Preview of spatial displacements Spatial Kinematics. Constraint. For Euclidean three space, are there ...... rotations that are not translations? Many. I Spherical I ... translations that are not rotations? Many. kinematics Euler’s theorem

I ... displacements that are both rotations and Spatial kinematics translations? One. Classifying displacements Chasles’s theorem I ... displacements that are neither? Twists! Screws Cones

I ... displacements that are not twists? No. Chasles’s Kinematic constraint theorem. Counting variables and equations General form Classifying constraints Mobility, Grübler Mobility and connectivity Grübler’s formula

Rotations Translations

Displacements

SE(3): Displacements of Euclidean three space Lecture 5 Chasles’s theorem Spatial Kinematics. Constraint. Theorem (2.7) Spherical kinematics Every spatial displacement is the composition of a Euler’s theorem rotation about some axis, and a translation along the Spatial kinematics Classifying displacements same axis. Chasles’s theorem Screws Cones Proof. Kinematic constraint Counting variables and equations I Assume arbitrary displacement D is given. General form Classifying constraints I Use theorem 2.2 to decompose D = R ◦ T . Mobility, Grübler Mobility and connectivity Grübler’s formula I Decompose T into components parallel to and perpendicular to axis of R: D = R ◦ T⊥ ◦ Tk.

I R ◦ T⊥ is planar — either a translation or a rotation. In either case the proof proceeds easily with D = (R ◦ T⊥) ◦ Tk as the desired decomposition. Lecture 5 Screws. Spatial Kinematics. Constraint.

Spherical Deﬁnition kinematics A screw is a line in space with an Euler’s theorem Spatial kinematics associated pitch, which is a ratio of Classifying displacements Chasles’s theorem linear to angular quantities. Screws Cones

Kinematic constraint Deﬁnition Counting variables and equations General form A twist is a screw plus a scalar Classifying constraints Mobility, Grübler magnitude, giving a rotation about the Mobility and connectivity screw axis plus a translation along the Grübler’s formula screw axis. The rotation angle is the twist magnitude, and the translation distance is the magnitude times the pitch. Thus the pitch is the ratio of translation to rotation. Lecture 5 Overview of the Screw idea Spatial Kinematics. Constraint.

Spherical kinematics Euler’s theorem

Spatial kinematics I It’s the geometrical object associated with various Classifying displacements Chasles’s theorem applications. Screws Cones

I A point is the geometrical object associated with a Kinematic constraint planar rotation—its rotation center. Counting variables and equations A line is the the geometrical object associated with a General form I Classifying constraints spatial rotation—the rotation axis. Mobility, Grübler Mobility and connectivity Grübler’s formula I A screw is the geometrical object associated with a twist or a wrench. Lecture 5 Pitch Spatial Kinematics. Constraint.

Spherical kinematics Euler’s theorem

Spatial kinematics Classifying displacements Chasles’s theorem I For twists, we define pitch to be translation distance Screws over rotation angle. Cones Kinematic I When we define wrenches, we will define pitch to be constraint Counting variables and equations torque divided by force. General form Classifying constraints I So, cannot say in the abstract, angular / linear, or Mobility, Grübler Mobility and connectivity linear / angular. Grübler’s formula Lecture 5 Analogous to centrodes ... Spatial Kinematics. Constraint.

Spherical kinematics Euler’s theorem

Spatial kinematics I On the sphere ... Classifying displacements Chasles’s theorem Screws I Plotting the instantaneous rotation Cones

axis in the fixed and moving Kinematic frames gives fixed and moving constraint Counting variables and cones. equations General form I In three space ... Classifying constraints Mobility, Grübler I Plotting the instantaneous screw Mobility and connectivity Grübler’s formula axis in the fixed and moving frames gives fixed and moving axodes. (How to draw pitch?) Lecture 5 Kinematic constraint Spatial Kinematics. Constraint. Motivation

Spherical kinematics Euler’s theorem

Why kinematic constraint? Spatial kinematics Classifying displacements Chasles’s theorem I Constraint is fundamental to manipulation. All the Screws Cones

good stuff happens at the boundaries, the nooks and Kinematic crannies, of the conﬁguration space. constraint Counting variables and equations I Simplest approach to manipulation: attach object General form Classifying constraints (grasp) to a programmable motion device (arm). Mobility, Grübler Mobility and connectivity Attachment = constraint. Grübler’s formula

I Many goals involve kinematic constraint: put something on a table, assemble something, etc.

I We use kinematic constraint to achieve precision. Lecture 5 Kinematic constraint: the main idea Spatial Kinematics. Constraint.

Spherical kinematics I In simple cases, freedoms and constraints are just a Euler’s theorem Spatial kinematics matter of counting unknowns and equations. Classifying displacements Chasles’s theorem Screws nominal DOFs Cones Kinematic −independent constraints constraint Counting variables and equations = DOFs General form Classifying constraints Mobility, Grübler I Things to worry about: Mobility and connectivity Grübler’s formula I If an equation reduces DOFs by 1, does an inequation reduce DOFs by 1/2? I Identifying dependencies and singular cases. I Constraints on velocity versus on conﬁguration. Lecture 5 Differential geometry Spatial Kinematics. Constraint.

Spherical kinematics Euler’s theorem

Spatial kinematics Classifying displacements Deﬁnition (Tangent space) Chasles’s theorem Screws Cones

Let Q be a conﬁguration Kinematic space, let q ∈ Q be a constraint Counting variables and equations conﬁguration. The tangent General form Classifying constraints space at qT qQ is the vector Mobility, Grübler Mobility and connectivity space comprising all velocity Grübler’s formula vectors at q. Lecture 5 Constraint in general Spatial Kinematics. Constraint.

Spherical kinematics Consider constraints of the form Euler’s theorem Spatial kinematics Classifying displacements f (q, q˙ , t) = 0 Chasles’s theorem Screws Cones

or Kinematic ˙ constraint f (q, q, t) ≥ 0 Counting variables and equations General form where Classifying constraints Mobility, Grübler Mobility and connectivity q ∈ Q conﬁguration space, e.g. (x, y, θ) Grübler’s formula

q˙ ∈ TqQ tangent space, e.g. (x˙ , y˙ , θ˙) t = time Lecture 5 Constraint: taxonomy and examples Spatial Kinematics. Constraint.

bilateral Spherical kinematics Expressed as an equation. Euler’s theorem Two sided. Spatial kinematics y Classifying displacements Chasles’s theorem y = 0 Screws Cones .x;y/ θ = 0 x Kinematic constraint Counting variables and equations unilateral General form Classifying constraints Expressed as an Mobility, Grübler 2 1 Mobility and connectivity inequation. One sided. y Grübler’s formula

y ≥ 0 (x,y) y + 2 sin θ ≥ 0 x y + 2 sin θ + cos θ ≥ 0 y + cos θ ≥ 0 Lecture 5 Constraint: taxonomy and examples Spatial Kinematics. Constraint.

scleronomic Spherical kinematics Independent of t. Euler’s theorem

Stationary. Spatial kinematics Classifying displacements rheonomic Chasles’s theorem y Screws Cones Depends on t. = 2 t (x,y) Kinematic constraint x x sin(2πt) − y cos(2πt) = 0 Counting variables and equations General form θ = 2πt Classifying constraints Mobility, Grübler Mobility and connectivity holonomic Grübler’s formula Independent of q˙ and y bilateral. (x,y) f (q, t) = 0 x nonholonomic Lecture 5 Constraint and kinematic mechanisms Spatial Kinematics. Constraint.

Spherical kinematics Euler’s theorem Link: a rigid body; Spatial kinematics Classifying displacements Joint: imposes one or Chasles’s theorem Planar Spherical Screws more constraints on 3freedoms 3 freedoms Cones the relative motion of Kinematic constraint two links; Counting variables and equations Cylindrical Revolute General form 1 freedom Kinematic 2freedoms Classifying constraints Mobility, Grübler mechanism: a bunch Mobility and connectivity of links joined by joints; Grübler’s formula Prismatic Helical lower pairs joints 1 freedom 1 freedom involving positive contact area. Two.

One. Two. One.

Lecture 5 Mobility and connectivity Spatial Kinematics. Constraint.

Spherical kinematics mobility of a mechanism: DOFs Euler’s theorem Spatial kinematics with one link ﬁxed. Classifying displacements Chasles’s theorem connectivity DOFs of one link Screws Cones relative to another. L4 Kinematic constraint What is the mobility of the ﬁve bar L3 Counting variables and equations linkage at right? General form Classifying constraints L5 Mobility, Grübler What is the connectivity of L2 L1 Mobility and connectivity Grübler’s formula I Link 1 relative to link two?

I Link 3 relative to link 1?

I Link 3 relative to link 4? One. Two. One.

Lecture 5 Mobility and connectivity Spatial Kinematics. Constraint.

Spherical kinematics mobility of a mechanism: DOFs Euler’s theorem Spatial kinematics with one link ﬁxed. Classifying displacements Chasles’s theorem connectivity DOFs of one link Screws Cones relative to another. L4 Kinematic constraint What is the mobility of the ﬁve bar L3 Counting variables and equations linkage at right? Two. General form Classifying constraints L5 Mobility, Grübler What is the connectivity of L2 L1 Mobility and connectivity Grübler’s formula I Link 1 relative to link two?

I Link 3 relative to link 1?

I Link 3 relative to link 4? Two. One.

Lecture 5 Mobility and connectivity Spatial Kinematics. Constraint.

Spherical kinematics mobility of a mechanism: DOFs Euler’s theorem Spatial kinematics with one link ﬁxed. Classifying displacements Chasles’s theorem connectivity DOFs of one link Screws Cones relative to another. L4 Kinematic constraint What is the mobility of the ﬁve bar L3 Counting variables and equations linkage at right? Two. General form Classifying constraints L5 Mobility, Grübler What is the connectivity of L2 L1 Mobility and connectivity Grübler’s formula I Link 1 relative to link two? One.

I Link 3 relative to link 1?

I Link 3 relative to link 4? One.

Lecture 5 Mobility and connectivity Spatial Kinematics. Constraint.

Spherical kinematics mobility of a mechanism: DOFs Euler’s theorem Spatial kinematics with one link ﬁxed. Classifying displacements Chasles’s theorem connectivity DOFs of one link Screws Cones relative to another. L4 Kinematic constraint What is the mobility of the ﬁve bar L3 Counting variables and equations linkage at right? Two. General form Classifying constraints L5 Mobility, Grübler What is the connectivity of L2 L1 Mobility and connectivity Grübler’s formula I Link 1 relative to link two? One.

I Link 3 relative to link 1? Two.

I Link 3 relative to link 4? Lecture 5 Mobility and connectivity Spatial Kinematics. Constraint.

Spherical kinematics mobility of a mechanism: DOFs Euler’s theorem Spatial kinematics with one link ﬁxed. Classifying displacements Chasles’s theorem connectivity DOFs of one link Screws Cones relative to another. L4 Kinematic constraint What is the mobility of the ﬁve bar L3 Counting variables and equations linkage at right? Two. General form Classifying constraints L5 Mobility, Grübler What is the connectivity of L2 L1 Mobility and connectivity Grübler’s formula I Link 1 relative to link two? One.

I Link 3 relative to link 1? Two.

I Link 3 relative to link 4? One. Lecture 5 Grübler’s formula Spatial Kinematics. Constraint. Given n links joined by g joints, Spherical with ui constraints and fi freedoms at joint i. (Note that kinematics u + f = 6.) Euler’s theorem i i Spatial kinematics Classifying displacements Assume one link is ﬁxed and constraints are all Chasles’s theorem Screws independent. Cones

Kinematic The mobility M is constraint Counting variables and X equations M = 6(n − 1) − ui General form Classifying constraints X Mobility, Grübler = 6(n − 1) − (6 − fi ) Mobility and connectivity Grübler’s formula X = 6(n − g − 1) + fi Or, for a planar mechanism: X M = 3(n − 1) − ui X = 3(n − g − 1) + fi Lecture 5 Grübler: special case for loops Spatial Kinematics. Constraint. The previous formula works (sort of) for all mechanisms. Spherical For loops there is a variant. kinematics Euler’s theorem

One loop: n = g, so Spatial kinematics Classifying displacements X Chasles’s theorem M = fi + 6(−1) Screws Cones Two loops: make a second loop by adding k links and Kinematic constraint k + 1 joints: Counting variables and X equations = + (− ) General form M fi 6 2 Classifying constraints Mobility, Grübler Every loop increases excess of joints over links by 1. For Mobility and connectivity Grübler’s formula l loops: X M = fi − 6l for a spatial linkage, and X M = fi − 3l for a planar linkage. 1 1 1

− 2 − 2 − 2

Lecture 5 Common sense Spatial Kinematics. Constraint. Example: what is the mobility of Watt’s Spherical linkage? kinematics Euler’s theorem

Planar Grübler’s formula: Spatial kinematics Classifying displacements X Chasles’s theorem M = 3(n − 1) − ui = Screws 5 Cones 3 X Kinematic 3 5 M = 3(n − g − 1) + fi = constraint 10 Counting variables and X equations M = fi − 3l = General form Classifying constraints Independent Mobility, Grübler Mobility and connectivity Spatial Grübler’s formula: constraints is Grübler’s formula X a very strong M = 6(n − 1) − ui = assumption. X M = 6(n − g − 1) + fi = X M = fi − 6l =

Why? 1 1

− 2 − 2 − 2

Lecture 5 Common sense Spatial Kinematics. Constraint. Example: what is the mobility of Watt’s Spherical linkage? kinematics Euler’s theorem

Planar Grübler’s formula: Spatial kinematics Classifying displacements X Chasles’s theorem M = 3(n − 1) − ui = 1 Screws 5 Cones 3 X Kinematic 3 5 M = 3(n − g − 1) + fi = constraint 10 Counting variables and X equations M = fi − 3l = General form Classifying constraints Independent Mobility, Grübler Mobility and connectivity Spatial Grübler’s formula: constraints is Grübler’s formula X a very strong M = 6(n − 1) − ui = assumption. X M = 6(n − g − 1) + fi = X M = fi − 6l =

Why? 1

− 2 − 2 − 2

Lecture 5 Common sense Spatial Kinematics. Constraint. Example: what is the mobility of Watt’s Spherical linkage? kinematics Euler’s theorem

Planar Grübler’s formula: Spatial kinematics Classifying displacements X Chasles’s theorem M = 3(n − 1) − ui = 1 Screws 5 Cones 3 X Kinematic 3 5 M = 3(n − g − 1) + fi = 1 constraint 10 Counting variables and X equations M = fi − 3l = General form Classifying constraints Independent Mobility, Grübler Mobility and connectivity Spatial Grübler’s formula: constraints is Grübler’s formula X a very strong M = 6(n − 1) − ui = assumption. X M = 6(n − g − 1) + fi = X M = fi − 6l =

Why? − 2 − 2 − 2

Lecture 5 Common sense Spatial Kinematics. Constraint. Example: what is the mobility of Watt’s Spherical linkage? kinematics Euler’s theorem

Planar Grübler’s formula: Spatial kinematics Classifying displacements X Chasles’s theorem M = 3(n − 1) − ui = 1 Screws 5 Cones 3 X Kinematic 3 5 M = 3(n − g − 1) + fi = 1 constraint 10 Counting variables and X equations M = fi − 3l = 1 General form Classifying constraints Independent Mobility, Grübler Mobility and connectivity Spatial Grübler’s formula: constraints is Grübler’s formula X a very strong M = 6(n − 1) − ui = assumption. X M = 6(n − g − 1) + fi = X M = fi − 6l =

Why? − 2 − 2

Lecture 5 Common sense Spatial Kinematics. Constraint. Example: what is the mobility of Watt’s Spherical linkage? kinematics Euler’s theorem

Planar Grübler’s formula: Spatial kinematics Classifying displacements X Chasles’s theorem M = 3(n − 1) − ui = 1 Screws 5 Cones 3 X Kinematic 3 5 M = 3(n − g − 1) + fi = 1 constraint 10 Counting variables and X equations M = fi − 3l = 1 General form Classifying constraints Independent Mobility, Grübler Mobility and connectivity Spatial Grübler’s formula: constraints is Grübler’s formula X a very strong M = 6(n − 1) − ui = − 2 assumption. X M = 6(n − g − 1) + fi = X M = fi − 6l =

Why? − 2

Lecture 5 Common sense Spatial Kinematics. Constraint. Example: what is the mobility of Watt’s Spherical linkage? kinematics Euler’s theorem

Planar Grübler’s formula: Spatial kinematics Classifying displacements X Chasles’s theorem M = 3(n − 1) − ui = 1 Screws 5 Cones 3 X Kinematic 3 5 M = 3(n − g − 1) + fi = 1 constraint 10 Counting variables and X equations M = fi − 3l = 1 General form Classifying constraints Independent Mobility, Grübler Mobility and connectivity Spatial Grübler’s formula: constraints is Grübler’s formula X a very strong M = 6(n − 1) − ui = − 2 assumption. X M = 6(n − g − 1) + fi = − 2 X M = fi − 6l =

Why? Lecture 5 Common sense Spatial Kinematics. Constraint. Example: what is the mobility of Watt’s Spherical linkage? kinematics Euler’s theorem

Planar Grübler’s formula: Spatial kinematics Classifying displacements X Chasles’s theorem M = 3(n − 1) − ui = 1 Screws 5 Cones 3 X Kinematic 3 5 M = 3(n − g − 1) + fi = 1 constraint 10 Counting variables and X equations M = fi − 3l = 1 General form Classifying constraints Independent Mobility, Grübler Mobility and connectivity Spatial Grübler’s formula: constraints is Grübler’s formula X a very strong M = 6(n − 1) − ui = − 2 assumption. X M = 6(n − g − 1) + fi = − 2 X M = fi − 6l = − 2

Why?