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 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 Many. 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? 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