DOCSLIB.ORG

Position and Orientation of Rigid Bodies

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Position and Orientation of Rigid Bodies Robotics 1 Position and orientation of rigid bodies Prof. Alessandro De Luca Robotics 1 1 Position and orientation right-handed orthogonal RFB rigid body Reference Frames A 3 • position: pAB (vector ! R ), B expressed in RF (use of coordinates I A RFA other than Cartesian is possible, e.g. pAB cylindrical or spherical) • orientation: orthonormal 3x3 matrix T -1 A B A (R = R " RB RA = I), with det = +1 A A A A RB = [ xB yB zB] • xA yA zA (xB yB zB) are unit vectors (with unitary norm) of frame RFA (RFB) A • components in RB are the direction cosines of the axes of RFB with respect to (w.r.t.) RFA Robotics 1 2 Rotation matrix T T T direction cosine of xA xB xA yB xA zB zB w.r.t. xA AR = y T x y T y y T z orthonormal, B A B A B A B T T T with det = +1 zA xB zA yB zA zB algebraic structure chain rule property of a group SO(3) (neutral element = I; k i k inverse element = RT) Ri ! Rj = Rj orientation of RFi orientation of RFj w.r.t. RFk w.r.t. RFk orientation of RFj w.r.t. RFi NOTE: in general, the product of rotation matrices does not commute! Robotics 1 3 Change of coordinates z0 0 px 0P = 0p = 1p 0x + 1p 0y + 1p 0z z1 • P y x 1 y 1 z 1 0p z 1 px y 1 0 0 0 1 = x1 y1 z1 py 1 pz RF1 y0 = 0R 1P RF0 1 x0 0 the rotation matrix R1 (i.e., the orientation of RF1 w.r.t. RF0) represents also the change of x1 coordinates of a vector from RF1 to RF0 Robotics 1 4 Ex: Orientation of frames in a plane (elementary rotation around z-axis) x = OB – xB = u cos # - v sin #$ y P • y = OC + Cy = u sin # + v cos #$ RFC z = w v # or… C u C 0 0 0 OP # RF0 xC yC zC O x B x cos # $-sin # $0 u u 0 OP y = sin # $ cos # 0 v = Rz(#) v z 0 0 1 w w R (-#) = R T(#) similarly: z z 1 0 0 cos # 0 sin # Rx(#) = 0 cos # - sin # Ry(#) = 0 1 0 0 sin # cos #$ - sin # 0 cos #$ Robotics 1 5 Ex: Rotation of a vector around z v’ x = |v| cos %$ y’ y = |v| sin %$ v y x’ = |v| cos (% + #) = |v| (cos % cos # - sin % sin #) # x’ = x cos # - y sin # %$ y’ = |v| sin (% + #) = |v| (sin % cos # + cos % sin #) x’ x O x’ = x sin # + y cos # z’ = z or… x’ cos # - sin # 0 x x …as y’ = sin # cos # 0 y = Rz(#) y before! z’ 0 0 1 z z Robotics 1 6 Equivalent interpretations of a rotation matrix the same rotation matrix, e.g., Rz(#), may represent: v’ RF C v • P RFC # # # RF RF0 0 the orientation of a rigid the change of coordinates the vector body with respect to a from RFC to RF0 rotation operator 0 C reference frame RF0 ex: P = Rz(#) P ex: v’ = Rz(#) v 0 0 0 ex: [ xc yc zc] = Rz(#) 0 the rotation matrix RC is an operator superposing frame RF0 to frame RFC Robotics 1 7 Composition of rotations 1 brings RF on RF 2 brings RF on RF brings RF0 on RF1 R2 1 2 R3 2 3 0 R1 • 3p p23 = 0 p = 0 01 p12 = 0 RF2 RF3 RF1 RF 0 a comment on computational complexity 63 products 0p = (0R 1R 2R ) 3p = 0R 3p 1 2 3 3 42 summations 0 0 1 2 3 27 products p = R1 ( R2 ( R3 p)) 18 summations 2p 1 Robotics 1 p 8 Axis/angle representation z 0 DATA r z • unit vector r (!r! = 1)$ r (positive if counterclockwise, as • P • # v seen from an “observer” placed like r) z1 y1 #$ DIRECT PROBLEM RF1 v’ ry find y0 RF0 0 0 0 R(#,r) = [ x1 y1 z1] rx RF is the result of rotating such that x0 1 RF by an angle around 0 # 0 1 0 0 the unit vector r P= R(#,r) P v’ = R(#,r) v x1 Robotics 1 9 Axis/angle: Direct problem z T 0 R(#,r) = C Rz(#) C 1 C concatenation of three rotations C-1 = CT r 3 C = n s r 2 z1 y1 Rz(#) RF1 after the first rotation y0 the z-axis coincides with r RF0 sequence of 3 rotations that n and s are orthogonal x 0 bring frame RF0 to superpose unit vectors such that with frame RF1 n " s = r, or nysz - synz = rx x1 nzsx - sznx = ry nxsy - sxny = rz Robotics 1 10 Axis/angle: Direct problem solution T R(#,r) = C Rz(#) C T c# - s# 0 n T R(#,r) = n s r s# c# 0 s 0 0 1 rT T T T T T = r r + (n n + s s ) c# + (s n - n s ) s# taking into account that T T T T C C = n n + s s + r r = I , and that 0 -r r z y skew-symmetric(r): T T s n - n s = 0 -r = S(r) x r " v = S(r)v = - S(v)r 0 depends only T T T R(#,r) = r r + (I - r r ) c# + S(r) s# = R (-#,r) = R(-#,-r) on r and # !! Robotics 1 11 Rodriguez formula v’ = R(#,r) v v’ = v cos # + (r " v) sin # + (1 - cos #)(rTv) r proof: T T R(#,r) v = (r r + (I - r r ) cos # + S(r) sin #)v T = r r v (1 - cos #) + v cos # + (r " v) sin # q.e.d. Robotics 1 14 Unit quaternion " to eliminate undetermined and singular cases arising in the axis/angle representation, one can use the unit quaternion representation Q = {-, .} = {cos(#/2), sin(#/2) r} a scalar 3-dim vector 2 2 " - + !.! = 1 (thus, “unit ...”) " (#, r) and (-#, -r) gives the same quaternion Q " the absence of rotation is associated to Q = {1, 0} " unit quaternions can be composed with special rules (in a similar way as in the product of rotation matrices) T Q 1*Q 2 = {-1-2 - .1 .2, -1.2 + -2.1 + .1".2} Robotics 1 20 Robotics 1 Minimal representations of orientation (Euler and roll-pitch-yaw angles) Homogeneous transformations Prof. Alessandro De Luca Robotics 1 1 “Minimal” representations rotation matrices: 9 elements inverse problem problem inverse - 3 orthogonality relationships - 3 unitary relationships = 3 independent variables direct problem problem direct sequence of 3 rotations around independent axes fixed (ai) or moving/current (a’i) axes 12 + 12 possible different sequences (e.g., XYX) actually, only 12 since {(a1 α1), (a2 α2), (a3 α3)} ≡ { (a’3 α3) , (a’2 α2), (a’1 α1)} Robotics 1 2 ZX’Z’’ Euler angles 1 z≡z’ z’ z’’ 2 y’’ RF” φ y’ RF’ θ y’ R (θ) = RF x’ θ 1 0 0 y 0 cos θ -sin θ x φ x’ x’≡x’’ 0 sin θ cos θ cos φ - sin φ 0 z’’ z’’’ Rz(φ) = sin φ cos φ 0 ≡ 3 y’’’ 0 0 1 y’’ ψ RF”’ cos ψ - sin ψ 0 R ( ) = z” ψ sin ψ cos ψ 0 x’’ x’’’ 0 0 1 ψ Robotics 1 3 ZX’Z’’ Euler angles direct problem: given φ , θ , ψ ; find R RZX’Z’’ (φ, θ, ψ) = RZ (φ) RX’ (θ) RZ’’ (ψ) order of definition cφ cψ - sφ cθ sψ - cφ sψ - sφ cθ cψ sφ sθ in concatenation = sφ cψ + cφ cθ sψ - sφ sψ + cφ cθ cψ - cφ sθ sθ sψ sθ cψ cθ given a vector v”’= (x”’,y”’,z”’) expressed in RF”’, its expression in the coordinates of RF is v = RZX’Z’’ (φ, θ, ψ) v”’ the orientation of RF”’ is the same that would be obtained with the sequence of rotations: ψ around z, θ around x (fixed), φ around z (fixed) Robotics 1 4 Roll-Pitch-Yaw angles ROLL PITCH 1 z’ 2 z z’’ z’ y’’ ψ T y’ C1RY(θ)C1 y’ θ with RY(θ) = θ y cos θ 0 sin θ y 0 1 0 x’ ψ x’’ - sin θ 0 cos θ x≡x’ YAW z 3 y’’’ 1 0 0 z’’ φ y’’ RX(ψ) = 0 cos ψ - sin ψ 0 sin ψ cos ψ T z’’’ C2RZ(φ)C2 cos φ - sin φ 0 with RZ(φ) = sin φ cos φ 0 0 0 1 x’’ φ x’’’ Robotics 1 6 Roll-Pitch-Yaw angles (fixed XYZ) direct problem: given ψ , θ , φ ; find R ⇐ note the order of products! RRPY (ψ, θ, φ) = RZ (φ) RY (θ) RX (ψ) order of definition cφ cθ cφ sθ sψ - sφ cψ cφ sθ cψ + sφ sψ = sφ cθ sφ sθ sψ + cφ cψ sφ sθ cψ - cφ sψ - sθ cθ sψ cθ cψ inverse problem: given R = {rij}; find ψ , θ , φ 2 2 2 2 2 r32 + r33 = c θ, r31 = -sθ ⇒ θ = ATAN2{-r31, ± √ r32 + r33 } 2 2 two symmetric values w.r.t. π/2 if r32 + r33 ≠ 0 (i.e., cθ ≠ 0) r32/cθ = sψ, r33/cθ = cψ ⇒ ψ = ATAN2{r32/cθ, r33/cθ} similarly... φ = ATAN2{ r21/cθ, r11/cθ } singularities for θ = ± π/2 Robotics 1 7 Homogeneous transformations P • Bp RF Ap B A ‘affine’ relationship pAB OB RFA A A A B p = pAB + RB p OA A A A B p RB pAB p linear A A B phom = = = TB phom relationship 1 0 0 0 1 1 vector in homogeneous 4x4 matrix of coordinates homogeneous transformation Robotics 1 9 Properties of T matrix describes the relation between reference frames (relative pose = position & orientation) transforms the representation of a position vector (applied vector from the origin of the frame) from a given frame to another frame it is a roto-translation operator on vectors in the three-dimensional space A -1 B it is always invertible ( TB) = TA A A B can be composed, i.e., TC = TB TC ← note: it does not commute! Robotics 1 10 Inverse of a homogeneous transformation A A A B B B B A A T A A T A p = pAB + RB p p = pBA + RA p = - RB pAB + RB p A A B B A T A T A RB pAB RA pBA RB - RB pAB = 0 0 0 1 0 0 0 1 0 0 0 1 A B A -1 TB TA ( TB) Robotics 1 11 Defining a robot task yE absolute definition of task task definition relative RFE to the robot end-effector zE RFB W W B E 2 • TT = TB TE TT 1 • RFT •3 known, once the robot direct kinematics of the is placed robot arm (function of q) RFW B W -1 W E -1 TE(q) = TB TT TT = cost Robotics 1 12 Final comments on T matrices they are the main tool for computing the direct kinematics of robot manipulators they are used in many application areas (in robotics and beyond) b in the positioning of a vision camera (matrix Tc with the extrinsic parameters of the camera posture) in computer graphics, for the real-time visualization of 3D solid objects when changing the observation point A A RB pAB A TB = αx αy αz σ all zero coefficients of scaling always unitary in robotics perspective coefficient in robotics deformation Robotics 1 13 Robotics 1 Direct kinematics Prof.
Load more

Recommended publications
  • Swimming in Spacetime: Motion by Cyclic Changes in Body Shape
    RESEARCH ARTICLE tation of the two disks can be accomplished without external torques, for example, by fixing Swimming in Spacetime: Motion the distance between the centers by a rigid circu- lar arc and then contracting a tension wire sym- by Cyclic Changes in Body Shape metrically attached to the outer edge of the two caps. Nevertheless, their contributions to the zˆ Jack Wisdom component of the angular momentum are parallel and add to give a nonzero angular momentum. Cyclic changes in the shape of a quasi-rigid body on a curved manifold can lead Other components of the angular momentum are to net translation and/or rotation of the body. The amount of translation zero. The total angular momentum of the system depends on the intrinsic curvature of the manifold. Presuming spacetime is a is zero, so the angular momentum due to twisting curved manifold as portrayed by general relativity, translation in space can be must be balanced by the angular momentum of accomplished simply by cyclic changes in the shape of a body, without any the motion of the system around the sphere. external forces. A net rotation of the system can be ac- complished by taking the internal configura- The motion of a swimmer at low Reynolds of cyclic changes in their shape. Then, presum- tion of the system through a cycle. A cycle number is determined by the geometry of the ing spacetime is a curved manifold as portrayed may be accomplished by increasing ␪ by ⌬␪ sequence of shapes that the swimmer assumes by general relativity, I show that net translations while holding ␾ fixed, then increasing ␾ by (1).
    [Show full text]
  • Foundations of Newtonian Dynamics: an Axiomatic Approach For
    Foundations of Newtonian Dynamics: 1 An Axiomatic Approach for the Thinking Student C. J. Papachristou 2 Department of Physical Sciences, Hellenic Naval Academy, Piraeus 18539, Greece Abstract. Despite its apparent simplicity, Newtonian mechanics contains conceptual subtleties that may cause some confusion to the deep-thinking student. These subtle- ties concern fundamental issues such as, e.g., the number of independent laws needed to formulate the theory, or, the distinction between genuine physical laws and deriva- tive theorems. This article attempts to clarify these issues for the benefit of the stu- dent by revisiting the foundations of Newtonian dynamics and by proposing a rigor- ous axiomatic approach to the subject. This theoretical scheme is built upon two fun- damental postulates, namely, conservation of momentum and superposition property for interactions. Newton’s laws, as well as all familiar theorems of mechanics, are shown to follow from these basic principles. 1. Introduction Teaching introductory mechanics can be a major challenge, especially in a class of students that are not willing to take anything for granted! The problem is that, even some of the most prestigious textbooks on the subject may leave the student with some degree of confusion, which manifests itself in questions like the following: • Is the law of inertia (Newton’s first law) a law of motion (of free bodies) or is it a statement of existence (of inertial reference frames)? • Are the first two of Newton’s laws independent of each other? It appears that
    [Show full text]
  • Molecular Symmetry
    Molecular Symmetry Symmetry helps us understand molecular structure, some chemical properties, and characteristics of physical properties (spectroscopy) – used with group theory to predict vibrational spectra for the identification of molecular shape, and as a tool for understanding electronic structure and bonding. Symmetrical : implies the species possesses a number of indistinguishable configurations. 1 Group Theory : mathematical treatment of symmetry. symmetry operation – an operation performed on an object which leaves it in a configuration that is indistinguishable from, and superimposable on, the original configuration. symmetry elements – the points, lines, or planes to which a symmetry operation is carried out. Element Operation Symbol Identity Identity E Symmetry plane Reflection in the plane σ Inversion center Inversion of a point x,y,z to -x,-y,-z i Proper axis Rotation by (360/n)° Cn 1. Rotation by (360/n)° Improper axis S 2. Reflection in plane perpendicular to rotation axis n Proper axes of rotation (C n) Rotation with respect to a line (axis of rotation). •Cn is a rotation of (360/n)°. •C2 = 180° rotation, C 3 = 120° rotation, C 4 = 90° rotation, C 5 = 72° rotation, C 6 = 60° rotation… •Each rotation brings you to an indistinguishable state from the original. However, rotation by 90° about the same axis does not give back the identical molecule. XeF 4 is square planar. Therefore H 2O does NOT possess It has four different C 2 axes. a C 4 symmetry axis. A C 4 axis out of the page is called the principle axis because it has the largest n . By convention, the principle axis is in the z-direction 2 3 Reflection through a planes of symmetry (mirror plane) If reflection of all parts of a molecule through a plane produced an indistinguishable configuration, the symmetry element is called a mirror plane or plane of symmetry .
    [Show full text]
  • (II) I Main Topics a Rotations Using a Stereonet B
    GG303 Lecture 11 9/4/01 1 ROTATIONS (II) I Main Topics A Rotations using a stereonet B Comments on rotations using stereonets and matrices C Uses of rotation in geology (and engineering) II II Rotations using a stereonet A Best uses 1 Rotation axis is vertical 2 Rotation axis is horizontal (e.g., to restore tilted beds) B Construction technique 1 Find orientation of rotation axis 2 Find angle of rotation and rotate pole to plane (or a linear feature) along a small circle perpendicular to the rotation axis. a For a horizontal rotation axis the small circle is vertical b For a vertical rotation axis the small circle is horizontal 3 WARNING: DON'T rotate a plane by rotating its dip direction vector; this doesn't work (see handout) IIIComments on rotations using stereonets and matrices Advantages Disadvantages Stereonets Good for visualization Relatively slow Can bring into field Need good stereonets, paper Matrices Speed and flexibility Computer really required to Good for multiple rotations cut down on errors A General comments about stereonets vs. rotation matrices 1 With stereonets, an object is usually considered to be rotated and the coordinate axes are held fixed. 2 With rotation matrices, an object is usually considered to be held fixed and the coordinate axes are rotated. B To return tilted bedding to horizontal choose a rotation axis that coincides with the direction of strike. The angle of rotation is the negative of the dip of the bedding (right-hand rule! ) if the bedding is to be rotated back to horizontal and positive if the axes are to be rotated to the plane of the tilted bedding.
    [Show full text]
  • Phoronomy: Space, Construction, and Mathematizing Motion Marius Stan
    To appear in Bennett McNulty (ed.), Kant’s Metaphysical Foundations of Natural Science: A Critical Guide. Cambridge University Press. Phoronomy: space, construction, and mathematizing motion Marius Stan With his chapter, Phoronomy, Kant defies even the seasoned interpreter of his philosophy of physics.1 Exegetes have given it little attention, and un- derstandably so: his aims are opaque, his turns in argument little motivated, and his context mysterious, which makes his project there look alienating. I seek to illuminate here some of the darker corners in that chapter. Specifi- cally, I aim to clarify three notions in it: his concepts of velocity, of compo- site motion, and of the construction required to compose motions. I defend three theses about Kant. 1) his choice of velocity concept is ul- timately insufficient. 2) he sided with the rationalist faction in the early- modern debate on directed quantities. 3) it remains an open question if his algebra of motion is a priori, though he believed it was. I begin in § 1 by explaining Kant’s notion of phoronomy and its argu- ment structure in his chapter. In § 2, I present four pictures of velocity cur- rent in Kant’s century, and I assess the one he chose. My § 3 is in three parts: a historical account of why algebra of motion became a topic of early modern debate; a synopsis of the two sides that emerged then; and a brief account of his contribution to the debate. Finally, § 4 assesses how general his account of composite motion is, and if it counts as a priori knowledge.
    [Show full text]
  • Euler Quaternions
    Four different ways to represent rotation my head is spinning... The space of rotations SO( 3) = {R ∈ R3×3 | RRT = I,det(R) = +1} Special orthogonal group(3): Why det( R ) = ± 1 ? − = − Rotations preserve distance: Rp1 Rp2 p1 p2 Rotations preserve orientation: ( ) × ( ) = ( × ) Rp1 Rp2 R p1 p2 The space of rotations SO( 3) = {R ∈ R3×3 | RRT = I,det(R) = +1} Special orthogonal group(3): Why it’s a group: • Closed under multiplication: if ∈ ( ) then ∈ ( ) R1, R2 SO 3 R1R2 SO 3 • Has an identity: ∃ ∈ ( ) = I SO 3 s.t. IR1 R1 • Has a unique inverse… • Is associative… Why orthogonal: • vectors in matrix are orthogonal Why it’s special: det( R ) = + 1 , NOT det(R) = ±1 Right hand coordinate system Possible rotation representations You need at least three numbers to represent an arbitrary rotation in SO(3) (Euler theorem). Some three-number representations: • ZYZ Euler angles • ZYX Euler angles (roll, pitch, yaw) • Axis angle One four-number representation: • quaternions ZYZ Euler Angles φ = θ rzyz ψ φ − φ cos sin 0 To get from A to B: φ = φ φ Rz ( ) sin cos 0 1. Rotate φ about z axis 0 0 1 θ θ 2. Then rotate θ about y axis cos 0 sin θ = ψ Ry ( ) 0 1 0 3. Then rotate about z axis − sinθ 0 cosθ ψ − ψ cos sin 0 ψ = ψ ψ Rz ( ) sin cos 0 0 0 1 ZYZ Euler Angles φ θ ψ Remember that R z ( ) R y ( ) R z ( ) encode the desired rotation in the pre- rotation reference frame: φ = pre−rotation Rz ( ) Rpost−rotation Therefore, the sequence of rotations is concatentated as follows: (φ θ ψ ) = φ θ ψ Rzyz , , Rz ( )Ry ( )Rz ( ) φ − φ θ θ ψ − ψ cos sin 0 cos 0 sin cos sin 0 (φ θ ψ ) = φ φ ψ ψ Rzyz , , sin cos 0 0 1 0 sin cos 0 0 0 1− sinθ 0 cosθ 0 0 1 − − − cφ cθ cψ sφ sψ cφ cθ sψ sφ cψ cφ sθ (φ θ ψ ) = + − + Rzyz , , sφ cθ cψ cφ sψ sφ cθ sψ cφ cψ sφ sθ − sθ cψ sθ sψ cθ ZYX Euler Angles (roll, pitch, yaw) φ − φ cos sin 0 To get from A to B: φ = φ φ Rz ( ) sin cos 0 1.
    [Show full text]
  • The Structuring of Pattern Repeats in the Paracas Necropolis Embroideries
    University of Nebraska - Lincoln DigitalCommons@University of Nebraska - Lincoln Textile Society of America Symposium Proceedings Textile Society of America 1988 Orientation And Symmetry: The Structuring Of Pattern Repeats In The Paracas Necropolis Embroideries Mary Frame University of British Columbia Follow this and additional works at: https://digitalcommons.unl.edu/tsaconf Part of the Art and Design Commons Frame, Mary, "Orientation And Symmetry: The Structuring Of Pattern Repeats In The Paracas Necropolis Embroideries" (1988). Textile Society of America Symposium Proceedings. 637. https://digitalcommons.unl.edu/tsaconf/637 This Article is brought to you for free and open access by the Textile Society of America at DigitalCommons@University of Nebraska - Lincoln. It has been accepted for inclusion in Textile Society of America Symposium Proceedings by an authorized administrator of DigitalCommons@University of Nebraska - Lincoln. 136 RESEARCH-IN-PROGRESS REPORT ORIENTATION AND SYMMETRY: THE STRUCTURING OF PATTERN REPEATS IN THE PARACAS NECROPOLIS EMBROIDERIES MARY FRAME RESEARCH ASSOCIATE, INSTITUTE OF ANDEAN STUDIES 4611 Marine Drive West Vancouver, B.C. V7W 2P1 The most extensive Peruvian fabric remains come from the archeological site of Paracas Necropolis on the South Coast of Peru. Preserved by the dry desert conditions, this cache of 429 mummy bundles, excavated in 1926-27, provides an unparal- leled opportunity for comparing the range and nature of varia- tions in similar fabrics which are securely related in time and space. The bundles are thought to span the time period from 500-200 B.C. The most numerous and notable fabrics are embroideries: garments that have been classified as mantles, tunics, wrap- around skirts, loincloths, turbans and ponchos.
    [Show full text]
  • Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
    Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation
    [Show full text]
  • The Problem of Exterior Orientation in Photogrammetry
    T he Problem of Exterior Orientation in Photogrammetry*t GEORGE H. ROSENFIELD, Data Reduction A nalyst, RCA Service Co., Air Force Missile Test Center, Patrick Air Force Base, Fla. ABSTRACT: Increased interest in Analytical Photogrammetry has focused at­ tention upon a chaotic situation which complicates research and communication between photogrammetrists. The designation of the orientation parameters in a different manner by almost every investigator has resulted in the need for a universal non-ambiguous system which depicts the actual physical orientation of the photograph. This paper presents a discussion of the basic concepts in­ volved in the various systems in use, and recommends the adoption of a standard system within the basic framework of the Stockholm resolution of July 1956, concerning the sign convention to be used in Photogrammetry. 1. INTRODUCTION CHAOTIC situation exists within the science of Photogrammetry which complicates A research and communication between photogrammetrists. When the theoretician studies a technical paper, he finds the work unnecessarily laborious until he has first become familiar with the coordinate system, definitions, and rotations employed by the writer. \i\Then the skilled instrument operator sits down to a newstereoplotting instrument he finds the orienta­ tion procedure unnecessarily difficult until he be­ comes familiar with the positive direction of the translational and rotational motions. Thus it is that almost every individual has devised his own unique system of designating the exterior orienta­ tion of the photograph. In order to alleviate these complications, the VIII International Congress for Photogrammetry at Stockholm in July 1956, accepted a resolution on the sign convention to be used in photogrammetry.
    [Show full text]
  • A Guided Tour to the Plane-Based Geometric Algebra PGA
    A Guided Tour to the Plane-Based Geometric Algebra PGA Leo Dorst University of Amsterdam Version 1.15{ July 6, 2020 Planes are the primitive elements for the constructions of objects and oper- ators in Euclidean geometry. Triangulated meshes are built from them, and reflections in multiple planes are a mathematically pure way to construct Euclidean motions. A geometric algebra based on planes is therefore a natural choice to unify objects and operators for Euclidean geometry. The usual claims of `com- pleteness' of the GA approach leads us to hope that it might contain, in a single framework, all representations ever designed for Euclidean geometry - including normal vectors, directions as points at infinity, Pl¨ucker coordinates for lines, quaternions as 3D rotations around the origin, and dual quaternions for rigid body motions; and even spinors. This text provides a guided tour to this algebra of planes PGA. It indeed shows how all such computationally efficient methods are incorporated and related. We will see how the PGA elements naturally group into blocks of four coordinates in an implementation, and how this more complete under- standing of the embedding suggests some handy choices to avoid extraneous computations. In the unified PGA framework, one never switches between efficient representations for subtasks, and this obviously saves any time spent on data conversions. Relative to other treatments of PGA, this text is rather light on the mathematics. Where you see careful derivations, they involve the aspects of orientation and magnitude. These features have been neglected by authors focussing on the mathematical beauty of the projective nature of the algebra.
    [Show full text]
  • Euclidean Space - Wikipedia, the Free Encyclopedia Page 1 of 5
    Euclidean space - Wikipedia, the free encyclopedia Page 1 of 5 Euclidean space From Wikipedia, the free encyclopedia In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions. The term “Euclidean” distinguishes these spaces from the curved spaces of non-Euclidean geometry and Einstein's general theory of relativity, and is named for the Greek mathematician Euclid of Alexandria. Classical Greek geometry defined the Euclidean plane and Euclidean three-dimensional space using certain postulates, while the other properties of these spaces were deduced as theorems. In modern mathematics, it is more common to define Euclidean space using Cartesian coordinates and the ideas of analytic geometry. This approach brings the tools of algebra and calculus to bear on questions of geometry, and Every point in three-dimensional has the advantage that it generalizes easily to Euclidean Euclidean space is determined by three spaces of more than three dimensions. coordinates. From the modern viewpoint, there is essentially only one Euclidean space of each dimension. In dimension one this is the real line; in dimension two it is the Cartesian plane; and in higher dimensions it is the real coordinate space with three or more real number coordinates. Thus a point in Euclidean space is a tuple of real numbers, and distances are defined using the Euclidean distance formula. Mathematicians often denote the n-dimensional Euclidean space by , or sometimes if they wish to emphasize its Euclidean nature. Euclidean spaces have finite dimension. Contents 1 Intuitive overview 2 Real coordinate space 3 Euclidean structure 4 Topology of Euclidean space 5 Generalizations 6 See also 7 References Intuitive overview One way to think of the Euclidean plane is as a set of points satisfying certain relationships, expressible in terms of distance and angle.
    [Show full text]
  • Central Limit Theorems for the Brownian Motion on Large Unitary Groups
    CENTRAL LIMIT THEOREMS FOR THE BROWNIAN MOTION ON LARGE UNITARY GROUPS FLORENT BENAYCH-GEORGES Abstract. In this paper, we are concerned with the large n limit of the distributions of linear combinations of the entries of a Brownian motion on the group of n × n unitary matrices. We prove that the process of such a linear combination converges to a Gaussian one. Various scales of time and various initial distributions are considered, giving rise to various limit processes, related to the geometric construction of the unitary Brownian motion. As an application, we propose a very short proof of the asymptotic Gaussian feature of the entries of Haar distributed random unitary matrices, a result already proved by Diaconis et al. Abstract. Dans cet article, on consid`erela loi limite, lorsque n tend vers l’infini, de combinaisons lin´eairesdes coefficients d'un mouvement Brownien sur le groupe des matri- ces unitaires n × n. On prouve que le processus d'une telle combinaison lin´eaireconverge vers un processus gaussien. Diff´erentes ´echelles de temps et diff´erentes lois initiales sont consid´er´ees,donnant lieu `aplusieurs processus limites, li´es`ala construction g´eom´etrique du mouvement Brownien unitaire. En application, on propose une preuve tr`escourte du caract`ereasymptotiquement gaussien des coefficients d'une matrice unitaire distribu´ee selon la mesure de Haar, un r´esultatd´ej`aprouv´epar Diaconis et al. Introduction There is a natural definition of Brownian motion on any compact Lie group, whose distribution is sometimes called the heat kernel measure. Mainly due to its relations with the object from free probability theory called the free unitary Brownian motion and with the two-dimentional Yang-Mills theory, the Brownian motion on large unitary groups has appeared in several papers during the last decade.
    [Show full text]