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Direct & Inverse Kinematics1 Algorithmic Robotics and Motion Planning ( ) Instructor: Prof. Dan Halperin

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Direct & Inverse Kinematics2 Overview Kinematics Introduction to Protein Structure A kinematic View of Loop Closure

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Direct & Inverse Kinematics3 Overview Kinematics the science of motion that treats the subject without regard to the forces that cause it Introduction to Protein Structure A kinematic View of Loop Closure

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Direct & Inverse Kinematics4 Direct & inverse kinematics of manipulators What are we trying to do ? (direct) Go right !!! ???

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Direct & Inverse Kinematics5 Direct & inverse kinematics of manipulators What are we trying to do ? (inverse) Take the ball !!! ???

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Direct & Inverse Kinematics6 Spatial description and transformation Direct kinematics Inverse kinematics

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Direct & Inverse Kinematics7 Spatial description and transformation We need to be able to describe the position and the orientation of the robot’s parts Suppose there’s a universe coordinate system to which everything can be referenced.

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Direct & Inverse Kinematics8 Spatial description and transformation We need to be able to describe the position and the orientation of the robot’s parts (relative to U) What’s its position (“reference point”) ? What’s its orientation ?

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Direct & Inverse Kinematics9 Spatial description and transformation Spatial description Transformations Presentation of orientation Direct kinematics Inverse kinematics

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Direct & Inverse Kinematics10 Positions, orientations and frames The position of a point p relative to a coordinate system A ( A p):

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Direct & Inverse Kinematics11 Positions, orientations and frames The orientation of a body is described by a coordinate system B attached to the body, relative to A (a known coordinate system).

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Direct & Inverse Kinematics12 Positions, orientations and frames The orientation of a body is described by a coordinate system B attached to the body, relative to A (a known coordinate system). cosinus of the angle

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Direct & Inverse Kinematics13 Positions, orientations and frames A frame is a set of 4 vectors giving the position and orientation. Example: frame B

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Direct & Inverse Kinematics14 position Positions, orientations and frames Remember the robot’s part: orientation

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Direct & Inverse Kinematics15 Direct & Inverse Kinematics Spatial description and transformation Spatial description Transformations Presentation of orientation Direct kinematics Inverse kinematics

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Direct & Inverse Kinematics16 Mapping Until now, we say how to describe positions, orientations and frames. We need to be able to change descriptions from one frame to another: mapping. Mappings: –translated frames –rotated frames –general frames

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Direct & Inverse Kinematics17 Mappings involving translated frames Expressing a point B p in terms of frame {A}, when {A} has the same orientation as {B}:

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Direct & Inverse Kinematics18 Mappings involving rotated frames Expressing a vector B p in terms of frame {A}, when the origins of frames {A} and {B} are coincident:

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Direct & Inverse Kinematics19 A p‘s components are B p’s projections onto the unit directions of {A}. Remember the rotation matrix : it’s columns are the unit vectors of {B} expressed in {A}. Thus: Mappings involving rotated frames

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Direct & Inverse Kinematics20 Mappings involving rotated frames: example Given:frame {B} is rotated relative to frame {A} about Z by 30 degrees, and B P. Calc: A P

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Direct & Inverse Kinematics21 Mappings involving rotated frames: example Sol: exact computation !?

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Direct & Inverse Kinematics22 Mappings involving general frames {A} and {B} has different origins and orientations. Vector offset between origins: A p Borg {B} is rotated in respect to {A}:

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Direct & Inverse Kinematics23 Mappings involving general frames First, describe B p relative to a frame that has the same orientation of {A}, but whose origin coincides with the origin of {B} Then add A p Borg for the translation Thus:

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Direct & Inverse Kinematics24 Mappings involving general frames “Homogeneous transform”: A “transform” specifies a frame.

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Direct & Inverse Kinematics25 Multiplication of transforms Given C p. We want to find A p.

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Direct & Inverse Kinematics26 Compound transforms Given C p. We want to find A p. Frame {C} is known relative to frame {B}, and frame {B} is known relative to frame {A}.

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Direct & Inverse Kinematics27 Inverting a transform Frame {B} is known relative to frame {A} We want the description “frame {A} relative to frame {B}” Straightforward way: compute the inverse matrix (of a 4x4 matrix)

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Direct & Inverse Kinematics28 Inverting a transform Frame {B} is known relative to frame {A} We want the description “frame {A} relative to frame {B}” Better way: –Compute –Compute A P Borg :

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Direct & Inverse Kinematics29 Inverting a transform Frame {B} is known relative to frame {A} We want the description “frame {A} relative to frame {B}” Better way:

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Direct & Inverse Kinematics30 Transformations: example

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Direct & Inverse Kinematics31 Direct & Inverse Kinematics Spatial description and transformation Spatial description Transformations Presentation of orientation Direct kinematics Inverse kinematics

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Direct & Inverse Kinematics32 Presentation of Orientation Rotation matrices are useful as operator. Still, it’s “unnatural” to have to give elements of a matrix with orthonormal columns as input. There are several presentations which make that input process easier: –Fixed angles –Euler angles –Euler parameters –Quaternions

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Direct & Inverse Kinematics33 Fixed angles X-Y-Z fixed angles –Start with 2 frames: a fixed reference frame {A} and a coinciding frame {B} –First rotate {B} by γ about X A, then by β about Y A and finally by α about Z A. –The equivalent rotation matrix is:

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Direct & Inverse Kinematics34 Euler angles Z-Y-X Euler angles –Start with 2 frames: a moving reference (Euler angles) frame {B} and a coinciding fixed frame {A} –First rotate {B} by α about Z B, then by β about Y B and finally by γ about X B. –The equivalent rotation matrix is: The final result is the same as X-Y-Z fixed angles!!!

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Direct & Inverse Kinematics35 Fixed & Euler angles In general: 3 rotations taken about fixed axes (fixed angles) yield the same final orientation as the same 3 rotations taken in opposite order about the axes of the moving frame (Euler angles). There are other angle-set conventions: Z-Y-Z, etc. (both for fixed and moving reference frames).

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Direct & Inverse Kinematics36 Euler parameters Given an equivalent axis K=[K X K Y K Z ] T (a unit vector we want to rotate about) and an angle θ, the Euler parameters are defined as: The rotation matrix R ε is:

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Direct & Inverse Kinematics37 Quaternions Definition: a generalization of complex numbers, obtained by adding the elements i, j, and k to the real numbers, where i, j, and k satisfy: i 2 =j 2 =k 2 =ijk=-1. a+bi+cj+dk, with a,b,c and d real numbers Quaternion’s conjugate: a-bi-cj-dk Quaternions are associative, distributive and not commutative. Another representation: a+vector(b,c,d)

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Direct & Inverse Kinematics38 Quaternions A rotation about the unit vector K=[K X K Y K Z ] T by an angle θ, can be computed using the quaternion: p’, the rotation of point p(0,p X,p Y,p Z ), is given by: Rotations may be contatenated: q’s elements are the same as the Euler parameters!

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Direct & Inverse Kinematics39 Direct & Inverse Kinematics Spatial description and transformation Direct kinematics –Link description –Link-connection description –Affixing frames to links –Manipulator kinematics –Example Inverse kinematics

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Direct & Inverse Kinematics40 Link Description Think of the manipulator as a chain of bodies (links) connected by joints. We will consider manipulators constructed with joints of 1 degree of freedom (DOF): revolute and prismatic joints. The links are numbered from 0 (immobile base) to n (free end of the arm).

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Direct & Inverse Kinematics41 Link Description

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Direct & Inverse Kinematics42 Link Description

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Direct & Inverse Kinematics43 Link Description Joint axis i: the line about which link i rotates relative to link i-1 link i-1 can be specified by 2 numbers: link length a i-1 and link twist α i-1 Link length and twist are sufficient to define the relation between any 2 axes in space

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Direct & Inverse Kinematics44 Direct & Inverse Kinematics Spatial description and transformation Direct kinematics –Link description –Link-connection description –Affixing frames to links –Manipulator kinematics –Example Inverse kinematics

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Direct & Inverse Kinematics45 Link-connection description Neighboring links have a common axis 2 parameters define the link-connection: –Link offset d i : the distance along the common axis from one link to the next –Joint angle θ i : amount of rotation about the common axis The link offset d i is variable if joint i is prismatic The joint angle θ i is variable if the joint is revolute

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Direct & Inverse Kinematics46 Link-connection description

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Direct & Inverse Kinematics47 Link-connection description variable offset d i variable angle θi

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Direct & Inverse Kinematics48 First and last link in the chain The link length a i, and the link twist α i depend on the joint axis i and i+1. Convention: a 0 =a n =0 and α 0 =α n =0 Similar for the link offset d i and the joint angle θ i : if joint 1 is revolute, then d 1 =0. if joint 1 is prismatic, then θ 1 =0.

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Direct & Inverse Kinematics49 Denavit-Hartenberg notation Any robot can be described kinematically by 4 quantities for each link: –2 for the link –2 to describe the link’s connection For revolute joints, θ i is called the joint variable (the other 3 quantities are fixed). For prismatic joints, d i is the joint variable (the other 3 quantities are fixed).

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Direct & Inverse Kinematics50 Denavit-Hartenberg notation The definition of mechanics by means of these quantities is called the Denavit- Hartenberg notation.

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Direct & Inverse Kinematics51 Direct & Inverse Kinematics Spatial description and transformation Direct kinematics –Link description –Link-connection description –Affixing frames to links –Manipulator kinematics –Example Inverse kinematics

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Direct & Inverse Kinematics52 Affixing frames to links We define a frame attached to each link: frame {i} is attached rigidly to link {i} Convention: –the origin is located where the “link length line” a i intersects the joint axis –the Z axis is coincident with the joint axis –the X axis points along a i, to the direction from joint i to joint i+1 –The Y axis is formed by the right-hand rule

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Direct & Inverse Kinematics53 Affixing frames to links

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Direct & Inverse Kinematics54 First and last link in the chain Frame {0} is the immobile base (link 0) of the robot. Thus a 0 =0 and α 0 =0. If joint 1 is revolute, then d 1 =0. If joint 1 is prismatic, then θ 1 =0. If joint n is revolute, then X n ’s direction is the same as X n-1 ’s (θ 1 =0), and {n}’s origin is the intersection of X n-1 and axis n when d n =0.

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Direct & Inverse Kinematics55 Example iα i-1 a i-1 didi θiθi 1000θ1θ1 20L1L1 0θ2θ2 30L2L2 0θ3θ3

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Direct & Inverse Kinematics56 Direct & Inverse Kinematics Spatial description and transformation Direct kinematics –Link description –Link-connection description –Affixing frames to links –Manipulator kinematics –Example Inverse kinematics

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Direct & Inverse Kinematics57 Manipulator kinematics We want to construct the transform that defines frame {i} relative to frame {i-1}, as a function of the four link parameters Each transform will be a function of only 1 joint variable Each link has his frame, thus the kinematics problem has been broken into n subproblems

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Direct & Inverse Kinematics58 Manipulator kinematics

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Direct & Inverse Kinematics59 Manipulator kinematics Each transform can be written as a combination of a translation and a rotation The single transformation that relates frame {n} to frame {0}:

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Direct & Inverse Kinematics60 Manipulator kinematics General form:

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Direct & Inverse Kinematics61 Frames with standard names

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Direct & Inverse Kinematics62 Direct & Inverse Kinematics Spatial description and transformation Direct kinematics –Link description –Link-connection description –Affixing frames to links –Manipulator kinematics –Example Inverse kinematics

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Direct & Inverse Kinematics63 Example: PUMA 560 PUMA 560 is a 6 DOFs industrial robot with all rotational joints (6R mechanism)

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Direct & Inverse Kinematics64 Example: PUMA 560 PUMA 560 is a 6 DOFs industrial robot with all rotational joints (6R mechanism)

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Direct & Inverse Kinematics65 Example: PUMA 560 Frame {0} and Frame {1} coincides when θ 1 =0. The joint axes Z 4, Z 5 and Z 6 (wrist’s joints) intersect at a common point. Z 4, Z 5 and Z 6 are mutually orthogonal.

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Direct & Inverse Kinematics66 Example: PUMA 560 Frames and link parameters:

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Direct & Inverse Kinematics67 Example: PUMA 560 Frames and link parameters:

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Direct & Inverse Kinematics68 Example: PUMA 560 Frames and link parameters: α i-1 a i-1 didi θiθi 1 000θ1θ1 2-90º00θ2θ2 3 0a2a2 d3d3 θ3θ3 4 a3a3 d4d4 θ4θ4 5 90º00θ5θ5 6-90º00θ6θ6

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Direct & Inverse Kinematics69 Example: PUMA 560 Link transformations:

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Direct & Inverse Kinematics70 Example: PUMA 560 The kinematics equations of the PUMA 560:

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Direct & Inverse Kinematics71 Direct & Inverse Kinematics Spatial description and transformation Direct kinematics Inverse kinematics –About the problem –Method of solution –Example

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Direct & Inverse Kinematics72 About the problem We now want to compute the set of joint variables, given the desired position and orientation of the tool relative to the station. The problem is non-linear: given, find the values of θ 1,…,θ n. These equations can be non-linear, transcendental.

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Direct & Inverse Kinematics73 About the problem Consider the PUMA 560: –Given the matrice, solve PUMA 560’s kinematics equations for the joint angles θ 1 through θ 6. –We get 3 independent equations from the rotation-matrix part of and 3 equations from the position-vector part of. –Thus: 6 nonlinear, transcendental equations and 6 unknowns, and this for a “very simple” 6 DOFs manipulator!

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Direct & Inverse Kinematics74 About the problem We must ask the following questions: –Is there a solution? –Are there several solutions? –How to solve the problem?

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Direct & Inverse Kinematics75 Workspace If the goal (desired position and orientation) is in the reachable workspace, then there’s at least 1 solution. The reachable workspace is dependent on the manipulator. There might be several solutions. We won’t cover the following considerations: obstacles, limits on joint ranges,...

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Direct & Inverse Kinematics76 Multiple solutions

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Direct & Inverse Kinematics77 Direct & Inverse Kinematics Spatial description and transformation Direct kinematics Inverse kinematics –About the problem –Method of solution –Example

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Direct & Inverse Kinematics78 Method of solution There’s no general algorithm Consider a manipulator as “solvable” if it is possible to calculate all the solutions. Manipulator solution strategies might be split into 2 classes: closed-form and numerical solutions (numerical solutions won’t be covered – generally slower because of their iterative nature).

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Direct & Inverse Kinematics79 Method of solution closed-form solutions methods are solutions methods based on analytic expressions or on the solution of a polynomial of degree 4 or less. There’s a general numerical solution for which all “6 DOFs in a single chain” system with revolute and prismatic joints are solvable! Only on special cases can they be solved analytically.

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Direct & Inverse Kinematics80 Method of solution There are several closed-form solution strategies. We’ll discuss the following 2: –Algebraic solution by reduction to polynomial –Pieper’s criteria We’ll also discuss a heuristically solution strategy: –Cyclic Coordinate Descent (CCD)

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Direct & Inverse Kinematics81 Algebraic solution by reduction to polynomial Substitution: Advantage: –the substitution yields an expression in terms of variable u i instead of sin θ i and cos θ i. Once the solutions for u i are found, θ i =2tan -1 (solutions-of-u i ).

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Direct & Inverse Kinematics82 Pieper’s criteria There’s a closed-form solution for 6 DOFs manipulators (with prismatic and/or revolute joints configurations) in which 3 consecutive axes intersect in 1 point. Almost every 6 DOFs manipulator built today respect Pieper’s criteria. For ex: PUMA 560’s joint axes 4, 5 and 6.

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Direct & Inverse Kinematics83 Cyclic Coordinate Descent Algorithm: –adjusting one DOF at a time (iterative) to minimize tool’s distance to the goal –starts at the last link and works backwards, adjusting each joint along the way –repeat the whole set until “satisfied” or maximum nr. of sets reached Each step results in one equation with one unknown for each degree.

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Direct & Inverse Kinematics84 Cyclic Coordinate Descent Adjusting one link at the time Tool’s current position Goal’s position minimize Joint to move

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Direct & Inverse Kinematics85 Cyclic Coordinate Descent Adjusting one link at the time

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Direct & Inverse Kinematics86 Cyclic Coordinate Descent starts at the last link, adjusting each joint along the way repeat until “satisfied”

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Direct & Inverse Kinematics87 Cyclic Coordinate Descent Advantages: –Allow constraints to be placed (at each step) –Free of singularities –Degree independent –Computationally inexpensive (fast) –Simple to implement Disadvantage: –Might not find a solution

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Direct & Inverse Kinematics88 Direct & Inverse Kinematics Spatial description and transformation Direct kinematics Inverse kinematics –About the problem –Method of solution –Example

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Direct & Inverse Kinematics89 Example: PUMA 560 We know the links transformations in θ i (1≤i≤6).

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Direct & Inverse Kinematics90 Example: PUMA 560 We know the links transformations in θ i (1≤i≤6). Thus, we know the transformation from {k} to {l} in θ i (1≤k,l≤6 and k≤i≤l w.l.o.g.).

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Direct & Inverse Kinematics91 Example: PUMA 560 We know the links transformations in θ i (1≤i≤6). Thus, we know the transformation from {k} to {l} in θ i (1≤k,l≤6 and k≤i≤l w.l.o.g.). We wish to solve:

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Direct & Inverse Kinematics92 Example: PUMA 560 We’ll search for a solvable equation iteratively: –multiply each side of the transform equation by an inverse to separate a variable –Search for a solvable equations

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Direct & Inverse Kinematics93 Example: PUMA 560 After calculating the right side, it can been seen there’s a solvable solution: And we get: -sin θ 1 p X + cos θ 1 p Y = d 3. a cosθ i + b sin θ i = c return 2 solutions:

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Direct & Inverse Kinematics94 Example: PUMA solutions were found for θ 1. For each one of them, we’ll continue to search for solvable equations…

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Direct & Inverse Kinematics95 Overview Kinematics Introduction to Protein Structure A kinematic View of Loop Closure

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Direct & Inverse Kinematics96 Overview Kinematics Introduction to Protein Structure What do proteins look like and why is it important? A kinematic View of Loop Closure

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Direct & Inverse Kinematics97 Proteins & Polypeptides Amino AcidsPolypeptides Proteins

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Direct & Inverse Kinematics98 Where are they? Proteins are very important molecules to all forms of life. They are one of the four basic building blocks of life: –carbohydrates (sugars) –lipids (fats) –nucleic acids (DNA and RNA) –proteins

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Direct & Inverse Kinematics99 Where are they? They serve all kinds of functions: –part of structural elements in a cell (small scale) –part of the fibers that make up your muscles (larger scale) –Enzymes –Antibodies –Hormones –...

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Direct & Inverse Kinematics100 What are they? Proteins are made up of a chain of amino acids linked together (peptide bonds). They may be seen as a chain (the backbone) with a lot of side chains (the residues). There are 20 of those proteins' building blocks (amino acids) What differs the amino acids is their residue.

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Direct & Inverse Kinematics101 Polypeptides and Proteins Definition: A polypeptide is a compound containing amino acid residues joined by peptide bonds. A protein may consist of one or more specific polypeptide chains, which generally undergo further structural configurations in the course of becoming functional proteins.

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Direct & Inverse Kinematics102 Polypeptides and Proteins

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Direct & Inverse Kinematics103 Structure Proteins have 4 increasingly complex levels of structure: –Primary: sequence of the amino acids –Secondary: common folding patterns seen in proteins, like the alpha helix or the beta sheet –Tertiary: three-dimensional structure of a single folded amino acid chain –Quaternary: the complete protein with all of the subunits together (only in proteins made up of more than one polypeptide chain)

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Direct & Inverse Kinematics104 Structure

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Direct & Inverse Kinematics105 3D structure's importance Example of the importance of the structure: Boiling an egg causes all the proteins it contains, the "white" of the egg, to change shape and hardens (solid). It still has the same primary structure as the original protein, but the tertiary structure (three dimensional shape) has been lost, and so have all the critical properties of the original protein!

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Direct & Inverse Kinematics106 Structure's importance Proteins can have very complex shapes, and the final form of the protein is essential to its intended function The process of changing the shape of a protein so that the function is lost is called denaturation.

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Direct & Inverse Kinematics107 Structure-based problems Docking: predicting whether (and how) one molecule will bind to another Folding: the process by which the chain of amino acids is modified to reach its final form. Loop Closure (see next)

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Direct & Inverse Kinematics108 Geometric Properties Simplified (fixed bond lengths and angles) Planarity of the Peptide Bond: A polypeptide chain (backbone) may be considered as a series of planes (peptide units) with two angles of rotation between each plane.

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Direct & Inverse Kinematics109 Geometric Properties Each peptide bond adds 2 DOFs,

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Direct & Inverse Kinematics110 Geometric Properties Each peptide bond adds 2 DOFs, and the residues too add some DOFs.

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Direct & Inverse Kinematics111 A Kinematic View of Loop Closure Evangelos A. Coutsias, Chaok Seok, Matthew P. Jacobson, Ken A. Dill

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Direct & Inverse Kinematics112 Overview Kinematics Introduction to Protein Structure A kinematic View of Loop Closure

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Direct & Inverse Kinematics113 Overview Kinematics Introduction to Protein Structure A kinematic View of Loop Closure redefining the loop closure problem as a “inverse kinematics” problem

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Direct & Inverse Kinematics114 Loop Closure The loop closure problem Tripeptide loop closure Generalizations of the method

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Direct & Inverse Kinematics115 The loop closure problem Definition: finding the ensemble of possible backbone structures of a chain segment of a protein molecule that is geometrically consistent with preceding and following part of the chain whose structures are given

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Direct & Inverse Kinematics116 The loop closure problem missing given a chain segment of the protein find possible backbone structure So that it is geometrically consistent with the preceding and following parts of the chain

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Direct & Inverse Kinematics117 The loop closure problem In his simplest form: given a molecular chain with inflexible bond length ad bond angles, find all possible arrangements with the property that all bond vectors are fixed in space except for a contiguous set and such that the changes are made in at most six intervening dihedral angles.

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Direct & Inverse Kinematics118 Protein bond with inflexible length inflexible bond angles The loop closure problem In his simplest form: find all possible angles

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Direct & Inverse Kinematics119 Loop Closure The loop closure problem Tripeptide loop closure Generalizations of the method

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Direct & Inverse Kinematics120 Tripeptide loop closure The six-torsion loop closure problem in simplified representation: fixed in space variables: τ i (i=1,2,3) constraints: θ i (i=1,2,3) τ1τ1 τ2τ2 τ3τ3 θ1θ1 θ2θ2 θ3θ3

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Direct & Inverse Kinematics121 Tripeptide loop closure The six-torsion loop closure problem in simplified representation. But there are only 3 rotation angles τ i ? σi=τi+δiσi=τi+δi

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Direct & Inverse Kinematics122 Affixing frames

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Direct & Inverse Kinematics123 Affixing frames τ i angle is the rotation angle of r i τ about Z i

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Direct & Inverse Kinematics124 Affixing frames σ i angle is the rotation angle of r i σ about Z i

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Direct & Inverse Kinematics125 Constraints We’ve defined the frames so that they’ll be “easy to use”: –τ i is defined by r i τ –σ i is defined by r i σ The θ i angle constraints can be expressed in terms of r i τ and r i σ : r i τ. r i σ = cos θ i

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Direct & Inverse Kinematics126 Constraints r i τ. r i σ = cos θ i θiθi

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Direct & Inverse Kinematics127 Constraints τ i angle is the rotation angle of r i τ about Z i X i sin η i Z i cos η i

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Direct & Inverse Kinematics128 Constraints

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Direct & Inverse Kinematics129 Constraints Same for σ i angle: is the rotation angle of r i σ about Z i

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Direct & Inverse Kinematics130 Constraints

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Direct & Inverse Kinematics131 Constraints Those equations describe the rotation of the C αi-1 -N i and C αi -C i bonds about the virtual bonds C αi-1 -C αi and C αi -C αi+1 respectively

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Direct & Inverse Kinematics132 Solving the equations The angles α i, η i, ξ i, θ i and δ i depend on the bonds, which inflexible structure is known. Thus, they’re known constants. Convert the equations to polynomial variables w i and u i :

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Direct & Inverse Kinematics133 Solving the equations The equations becomes:

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Direct & Inverse Kinematics134 Solving the equations Eliminating w i :

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Direct & Inverse Kinematics135 Solving the equations There are now 3 equations, quadratic in 2 variable (u i and u i+1 ) Eliminating u 1 and then u 2 results in a degree 16 polynomial in u 3 There might be up to 16 solutions, even if at most 10 real solutions has been found in the article’s research

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Direct & Inverse Kinematics136 Loop Closure The loop closure problem Tripeptide loop closure Generalizations of the method

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Direct & Inverse Kinematics137 Generalizations of the method There are no assumptions about the intervening structures. Therefore, the algorithm can be applied to moves involving arbitrary triads of C α atoms.

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Direct & Inverse Kinematics138 Generalizations of the method Finding alternative local structures when an arbitrary dihedral angle is changed

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Direct & Inverse Kinematics139 Generalizations of the method The constraints form stay unchanged: But, C α1 -C α2, η1 and ξ2 changed. As a result, Z i (i=1,2) and α i (i=1,2,3) changed too. Still, the equations can be derived with the changed parameters

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Direct & Inverse Kinematics140 References Introduction to Robotics : Mechanics and Control (3rd Edition) John J. Craig Kinematic View of Loop Closure Evangelos A. Coutsias, Chaok Seok, Matthew P. Jacobson, Ken A. Dill Cyclic Coordinate Descent: A Robotics Algorithm for Protein Loop Closure Adrian A. Canutescu, Roland L. Dunbrack Jr.

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Direct & Inverse Kinematics141 Thank you

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