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## Introduction to Robotics Lecture II

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**Introduction to RoboticsLecture II**Alfred Bruckstein Yaniv Altshuler**Denavit-Hartenberg**• Specialized description of articulated figures • Each joint has only one degree of freedom • rotate around its z-axis • translate along its z-axis**Denavit-Hartenberg**• One degree of freedom : very compact notation • Only fourparameters to describe a relation between two links : • link length • link twist • link offset • link rotation**Denavit-Hartenberg**• Link length ai • The perpendicular distance between the axes of jointi and jointi+1**Denavit-Hartenberg**• Link twist αi • The angle between the axes of jointi and jointi+1 • Angle around xi-axis**Denavit-Hartenberg**• Link offset di • The distance between the origins of the coordinate frames attached to jointi and jointi+1 • Measured along the axis of jointi**Denavit-Hartenberg**• Link rotation (joint angle) φi • The angle between the link lenghts αi-1 and αi • Angle around zi-axis**Denavit-Hartenberg**• How to compute the parameters to describe an articulated figure : • Compute the link vector ai and the link length • Attach coordinate frames to the joint axes • Compute the link twist αi**Denavit-Hartenberg**• Compute the link offset di • Compute the joint angle φi • Compute the transformation (i-1)Ti which transforms entities from linki to linki-1**Denavit-Hartenberg**• Let’s do it step by step • Compute the link vector ai and the link length • Attach coordinate frames to the joint axes • Compute the link twist αi • Compute the link offset di • Compute the joint angle φi • Compute the transformation (i-1)Ti which transforms entities from linki to linki-1**Denavit-Hartenberg**The link length ai is the shortest distance between the joint axes jointi and jointi+1. Let the joint axes be given by the expression : Where pi is a point on axis of jointi and ui is one of its direction vectors (analogous for jointi+1).**Denavit-Hartenberg**• There are three methods to compute the link vector ai and the link length**Denavit-Hartenberg**• Method 1 : The Pseudo-naive approach The shortest distance aiis the length of the vector connecting the two axes, and perpendicular to both of them. Which can be expressed :**Denavit-Hartenberg**Let’s find the points oi and oai where this distance exists.**We can go some distance s from pi along axisi, and then the**distance ai along the unit vector and finally some distance t along axisi+1 to arrive at point pi+1. Denavit-Hartenberg**Denavit-Hartenberg**Multiplying respectively by ui and ui+1, we obtain the two following equations:**Denavit-Hartenberg**Solution :**Denavit-Hartenberg**Finally, using and we obtain :**Denavit-Hartenberg**• Method 2 : The Geometric approach The vector ui x ui+1 gives the perpendicular vector to both axes. Let’s find out where it is located on the joint axes. We can go some distance s from point pi along the axisi, and then go some distance k along ui x ui+1. Finally go some distance t along the axisi+1 to arrive at point pi+1.**Denavit-Hartenberg**We obtain the equation : There are three unknowns.**Denavit-Hartenberg**Let’s first eliminate the unknown k from the equation : by multiplying by ui:**Denavit-Hartenberg**Let’s first eliminate the unknown k from the equation : by multiplying by ui+1:**Denavit-Hartenberg**Now we shall eliminate the s and t from the equation : by multiplying by ui x ui+1:**Denavit-Hartenberg**We have obtained a system of three equations in the unknowns s, t, k :**Denavit-Hartenberg**From , it can be seen that the shortest distance between jointi and jointi+1 is given by the vector : Where**Denavit-Hartenberg**From and , we can compute s and t :**Denavit-Hartenberg**Finally, using and we obtain :**Denavit-Hartenberg**• Method 3 : The Analytic approach The distance between two arbitrary points located on the joint axes jointi and jointi+1 is :**Denavit-Hartenberg**The link length of linki, ai, is the minimum distance between the joint axes :**Denavit-Hartenberg**A necessary condition is :**Denavit-Hartenberg**Which is equivalent to their numerators being equal to 0 :**Denavit-Hartenberg**Rewriting this system yields :**Denavit-Hartenberg**Whose solution are :**Denavit-Hartenberg**Finally, using and we obtain :**Denavit-Hartenberg**oi and oaiare the closest points on the axes of jointi and jointi+1. We deduce that the link vector aiand the link length ai :**Denavit-Hartenberg**The link vector ai:**Denavit-Hartenberg**Calculating the scalar products and, both equal to 0, proves that the vector ai is perpendicular to both axes of jointi and jointi+1**Denavit-Hartenberg**• Three methods • How do we actually compute ai and ||ai||2 ?**Denavit-Hartenberg**The link vector ai is perpendicular to both of the axes of jointi and jointi+1. The unit vector : is parallel to the link vector ai.**Denavit-Hartenberg**Given two points pi and pi+1on the axes of jointi and jointi+1, the link length can be computed as : And the link vector :**Denavit-Hartenberg**• Special cases : • The joint axes intersect • The shortest distance ai is equal to zero • The link vector is the null vector**Denavit-Hartenberg**• The joint axes are parallel • There is no unique shortest distance oi can be chosen arbitrarily, so we should chose values that offset the most of Denavit-Hartenberg parameters**Denavit-Hartenberg**• The first joint • There is no link preceding it • We use a base link : link0 • Its link frame should coincide with the link frame of link1 • Most of the Denavit-Hartenberg parameters will be equal to zero**Denavit-Hartenberg**• The last joint • There is no link succeding it • We use arbitrary values so that most of Denavit-Hartenberg parameters are equal to zero**Denavit-Hartenberg**• Compute the link vector ai and the link lenght • Attach coordinate frames to the joint axes • Compute the link twist αi • Compute the link offset di • Compute the joint angle φi • Compute the transformation (i-1)Ti which transforms entities from linki to linki-1**Denavit-Hartenberg**• Identify the joint axes • Identify the common perpendiculars of successive joint axes • Attach coordinate frames to each joint axes**Denavit-Hartenberg**Identifying the joint axes**Remember, is the point where the shortest distance to**jointi+1 exists Denavit-Hartenberg Identifying the common perpendiculars**the origin**Denavit-Hartenberg Attaching the frames