# Direct & Inverse Kinematics

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

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

Direct & Inverse Kinematics
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 Direct & Inverse Kinematics

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

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

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

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. Direct & Inverse Kinematics

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 ? Direct & Inverse Kinematics

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

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

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). Direct & Inverse Kinematics

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 Direct & Inverse Kinematics

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

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

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

Direct & Inverse Kinematics
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 Direct & Inverse Kinematics

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

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

Mappings involving rotated frames
Ap‘s components are Bp’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: Direct & Inverse Kinematics

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

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

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

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

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

Multiplication of transforms
Given Cp. We want to find Ap. Direct & Inverse Kinematics

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

Direct & Inverse Kinematics
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) Direct & Inverse Kinematics

Direct & Inverse Kinematics
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 APBorg: Direct & Inverse Kinematics

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

Transformations: example
Direct & Inverse Kinematics

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

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 Direct & Inverse Kinematics

Direct & Inverse Kinematics
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 XA, then by β about YA and finally by α about ZA. The equivalent rotation matrix is: Direct & Inverse Kinematics

Direct & Inverse Kinematics
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 ZB, then by β about YB and finally by γ about XB. The equivalent rotation matrix is: The final result is the same as X-Y-Z fixed angles!!! Direct & Inverse Kinematics

Direct & Inverse Kinematics
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). Direct & Inverse Kinematics

Direct & Inverse Kinematics
Euler parameters Given an equivalent axis K=[KX KY KZ]T (a unit vector we want to rotate about) and an angle θ, the Euler parameters are defined as: The rotation matrix Rε is: Direct & Inverse Kinematics

Direct & Inverse Kinematics
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: i2=j2=k2=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) Direct & Inverse Kinematics

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

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

Direct & Inverse Kinematics
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). Direct & Inverse Kinematics

Direct & Inverse Kinematics
Link Description Direct & Inverse Kinematics

Direct & Inverse Kinematics
Link Description Direct & Inverse Kinematics

Direct & Inverse Kinematics
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 ai-1 and link twist αi-1 Link length and twist are sufficient to define the relation between any 2 axes in space Direct & Inverse Kinematics

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

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

Direct & Inverse Kinematics

variable offset di variable angle θi Direct & Inverse Kinematics

First and last link in the chain
The link length ai, and the link twist αi depend on the joint axis i and i+1. Convention: a0=an=0 and α0=αn=0 Similar for the link offset di and the joint angle θi : if joint 1 is revolute, then d1=0. if joint 1 is prismatic, then θ1=0. Direct & Inverse Kinematics

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, θi is called the joint variable (the other 3 quantities are fixed). For prismatic joints, di is the joint variable (the other 3 quantities are fixed). Direct & Inverse Kinematics

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

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

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” ai intersects the joint axis the Z axis is coincident with the joint axis the X axis points along ai, to the direction from joint i to joint i+1 The Y axis is formed by the right-hand rule Direct & Inverse Kinematics

Direct & Inverse Kinematics

First and last link in the chain
Frame {0} is the immobile base (link 0) of the robot. Thus a0=0 and α0=0. If joint 1 is revolute, then d1=0. If joint 1 is prismatic, then θ1=0. If joint n is revolute, then Xn’s direction is the same as Xn-1’s (θ1=0), and {n}’s origin is the intersection of Xn-1 and axis n when dn=0. Direct & Inverse Kinematics

Direct & Inverse Kinematics
Example i αi-1 ai-1 di θi 1 θ1 2 L1 θ2 3 L2 θ3 Direct & Inverse Kinematics

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

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 Direct & Inverse Kinematics

Manipulator kinematics
Direct & Inverse Kinematics

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}: Direct & Inverse Kinematics

Manipulator kinematics
General form: Direct & Inverse Kinematics

Frames with standard names
Direct & Inverse Kinematics

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

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

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

Direct & Inverse Kinematics
Example: PUMA 560 Frame {0} and Frame {1} coincides when θ1=0. The joint axes Z4, Z5 and Z6 (wrist’s joints) intersect at a common point. Z4, Z5 and Z6 are mutually orthogonal. Direct & Inverse Kinematics

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

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

Direct & Inverse Kinematics
Example: PUMA 560 Frames and link parameters: αi-1 ai-1 di θi 1 θ1 2 -90º θ2 3 a2 d3 θ3 4 a3 d4 θ4 5 90º θ5 6 θ6 Direct & Inverse Kinematics

Direct & Inverse Kinematics
Example: PUMA 560 Link transformations: Direct & Inverse Kinematics

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

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

Direct & Inverse Kinematics
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 θ1,…,θn . These equations can be non-linear, transcendental. Direct & Inverse Kinematics

Direct & Inverse Kinematics
About the problem Consider the PUMA 560: Given the matrice , solve PUMA 560’s kinematics equations for the joint angles θ1 through θ6. 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! Direct & Inverse Kinematics

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

Direct & Inverse Kinematics
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,... Direct & Inverse Kinematics

Direct & Inverse Kinematics
Multiple solutions Direct & Inverse Kinematics

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

Direct & Inverse Kinematics
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). Direct & Inverse Kinematics

Direct & Inverse Kinematics
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. Direct & Inverse Kinematics

Direct & Inverse Kinematics
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) Direct & Inverse Kinematics

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

Direct & Inverse Kinematics
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. Direct & Inverse Kinematics

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. Direct & Inverse Kinematics

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

Cyclic Coordinate Descent

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

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 Direct & Inverse Kinematics

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

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

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

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

Direct & Inverse Kinematics
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 Direct & Inverse Kinematics

Direct & Inverse Kinematics
Example: PUMA 560 After calculating the right side, it can been seen there’s a solvable solution: And we get: -sin θ1 pX + cos θ1 pY = d3 . a cosθi + b sin θi = c return 2 solutions: Direct & Inverse Kinematics

Direct & Inverse Kinematics
Example: PUMA 560 2 solutions were found for θ1. For each one of them, we’ll continue to search for solvable equations… Direct & Inverse Kinematics

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

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

Proteins & Polypeptides
Amino Acids Polypeptides Proteins Direct & Inverse Kinematics

Direct & Inverse Kinematics
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 Direct & Inverse Kinematics

Direct & Inverse Kinematics
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 ... Direct & Inverse Kinematics

Direct & Inverse Kinematics
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. Direct & Inverse Kinematics

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. Direct & Inverse Kinematics

Polypeptides and Proteins
Direct & Inverse Kinematics

Direct & Inverse Kinematics
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) Direct & Inverse Kinematics

Direct & Inverse Kinematics
Structure Direct & Inverse Kinematics

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! Direct & Inverse Kinematics

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. Direct & Inverse Kinematics

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) Direct & Inverse Kinematics

Direct & Inverse Kinematics
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. Direct & Inverse Kinematics

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

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

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

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

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

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

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 Direct & Inverse Kinematics

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

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. Direct & Inverse Kinematics

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

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

Tripeptide loop closure
The six-torsion loop closure problem in simplified representation: variables: τi (i=1,2,3) constraints: θi (i=1,2,3) τ1 τ2 τ3 θ1 θ2 θ3 fixed in space Direct & Inverse Kinematics

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

Direct & Inverse Kinematics
Affixing frames Direct & Inverse Kinematics

Direct & Inverse Kinematics
Affixing frames τi angle is the rotation angle of riτ about Zi Direct & Inverse Kinematics

Direct & Inverse Kinematics
Affixing frames σi angle is the rotation angle of ri σ about Zi Direct & Inverse Kinematics

Direct & Inverse Kinematics
Constraints We’ve defined the frames so that they’ll be “easy to use”: τi is defined by riτ σi is defined by ri σ The θi angle constraints can be expressed in terms of riτ and ri σ: riτ . ri σ = cos θi Direct & Inverse Kinematics

Direct & Inverse Kinematics
Constraints riτ . ri σ = cos θi θi Direct & Inverse Kinematics

Direct & Inverse Kinematics
Constraints τi angle is the rotation angle of riτ about Zi Xi sin ηi Zi cos ηi Direct & Inverse Kinematics

Direct & Inverse Kinematics
Constraints Direct & Inverse Kinematics

Direct & Inverse Kinematics
Constraints Same for σi angle: is the rotation angle of riσ about Zi Direct & Inverse Kinematics

Direct & Inverse Kinematics
Constraints Direct & Inverse Kinematics

Direct & Inverse Kinematics
Constraints Those equations describe the rotation of the Cαi-1-Ni and Cαi-Ci bonds about the virtual bonds Cαi-1-Cαi and Cαi-Cαi+1 respectively Direct & Inverse Kinematics

Direct & Inverse Kinematics
Solving the equations The angles αi, ηi, ξi, θi and δi depend on the bonds, which inflexible structure is known. Thus, they’re known constants. Convert the equations to polynomial variables wi and ui: Direct & Inverse Kinematics

Direct & Inverse Kinematics
Solving the equations The equations becomes: Direct & Inverse Kinematics

Direct & Inverse Kinematics
Solving the equations Eliminating wi : Direct & Inverse Kinematics

Direct & Inverse Kinematics
Solving the equations There are now 3 equations, quadratic in 2 variable (ui and ui+1) Eliminating u1 and then u2 results in a degree 16 polynomial in u3 There might be up to 16 solutions, even if at most 10 real solutions has been found in the article’s research Direct & Inverse Kinematics

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

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

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

Generalizations of the method
The constraints form stay unchanged: But, Cα1-Cα2, η1 and ξ2 changed. As a result, Zi (i=1,2) and αi (i=1,2,3) changed too. Still, the equations can be derived with the changed parameters Direct & Inverse Kinematics

Direct & Inverse Kinematics
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. Direct & Inverse Kinematics

Direct & Inverse Kinematics
Thank you Direct & Inverse Kinematics