1. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Kinematics Kinematics is the science of motion without regard to forces. We study the position, velocity, acceleration, jerk etc of objects Concerned with the location of Objects We will define coordinate systems or frames to define there location

2. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Definitions Velocity: The derivative of position with respect to time. Acceleration: The derivative of velocity with respect to time. Jerk: The derivative of acceleration with respect to time. Link: Nearly rigid structure between joints. Joint: Allow relative motion between links. Joint Angle: Measurement of the relative position of two links

3. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Definitions Continued Joint Space: Relative coordinates that are referenced to coordinate frames at the robot joints. Cartesian Space or Task Space. Global or base coordinate frame Jacobian: Specifies a mapping of Velocities in joint space to velocity in Cartesian or Task Space. Singularity: Region or point at which the Jacobian is singular.

4. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Mathematical Background

5. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Forward Kinematics Lets look at a simple link (1DOF)

6. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Forward Kinematics Want to know the end point of link in terms of X and Y We have R and Theta From Geometry we can determine the position: and then the velocity

7. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Two Link Example x Y x y    R1R1 R2R2

8. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Two Link Example X Y x y   

9. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Coordinate Frames R X0 Y0Y0 X1X1 Y1Y1

10. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Coordinate Frames Interested in Some Point described by a Vector R We need to express R which we know in Frame 1 in Frame 0 X0X0 Y0Y0 X1X1 Y1Y1 R 11  Equation is independent of values

11. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Coordinate Frames Now lets put it in matrix form So what if we want to map the other way? What is the inverse of T? Why?

12. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Coordinate Frames If we look at the columns and rows of T we see that they have a norm of one. Also if we take the dot product of the columns we find they are orthogonal to each other. So T is an ortho-normal Matrices. Thus its transpose is its inverse. This was a simple 2DOF example what about 3. If we project a Z axes out the plane generated by the X and Y axes, then a rotation around the Z axes will not affect the Z position of the vector R.

13. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Coordinate Frames The 3D transformation axes about the Z axes is: Similarly for rotations around the X or Y axes we get Shorthand notation is often used cos(  )=c  sin(  )=s 

14. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Coordinate Transformations For multiple transformation we simply multiply by more matrices. If we have multiple rotations about the same axes we can just add the angles of the matrices. Does this make sense. Substituting we get

15. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Coordinate Transformation So now we can express a vector in a reference frame rotated to an arbitrary angle in space. What if we want to express it in a translated frame

16. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Translating Coordinate Frames X0X0 Y0Y0 R X1X1 Y1Y1

17. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Translating Coordinate Frames X0X0 Y0Y0 R X1X1 Y1Y1 Now we can translate and rotate. Or in terms of vectors

18. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D.

19. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. What if we have a rotation and a translation First we can rotate the frames then we can translate X 0’ Y 0’ X1X1 Y1Y1 R 11  X0X0 Y0Y0

20. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Homogeneous Transformation Matrices 3x3 Rotation Matrix 3x1 Displacement Vector For a displacement and a rotation

21. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. 4x4 Homogeneous Matrix If we want to perform a rotation and a translation with one operation We can create a homogeneous Transformation Matrix

22. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Homogeneous Transformation Matrices What does a pure translation look like What does a pure rotation look like Multiply to move from one coordinate to another.

23. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. D-H Parameters Denavit-Hartenberg (D-H) are used to describe a robot link. One Coordinate System is created for each link. –Each Axes is orthogonal –Use the right hand rule Link are assumed to be rigid Four Parameters are used – d n Link Offset – a n Link Length (Common Normal Distance) –  n Joint Angle –  n Link Twist Angle

24. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Notation Coordinate systems are represented with brackets {B}, {0}, etc. Vectors –Lets Look at a Vector P Described in Frame A –Leading Subscript describes the frame in which the Vector is described or Referenced –Individual Elements of a vector are described by a trailing subscript Frame of Reference

25. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Matrix Notation New Frame of Reference Original Frame of Reference

26. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Homogenous Transformations Represent 3 Things Describe a Frame Map from one Frame to another Act as an Operator to move within a Frame

27. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Transforms Describe Frames Frames can be described by A Homogenous Transformation Matrices Description of Frame –Columns of are the Unit Vectors defining the directions of the principle axes of {B} in terms of {A} –Rows of are the Unit Vectors defining the directions of the principle axes of {A} in terms of {B} – is the location of the origin of {B} in terms or {A}

28. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Mapping Between Frames Maps vector from Frame {B} to Frame {A} will rotate a vector to project its components originally described in {B} in the {A} of Frame will translate the vector to adjust its origin from frame {B} to its new origin in {A}

29. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Acts as an Operator within a coordinate frame Operator will rotate and translate a vector Defines the angle to rotate about Defines the distance to translate Usually drop subscripts Operates on to create

30. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. anan Link Length (Common Normal) Measured from z n to z n+1 along x n The Common perpendicular is shortest distance between two lines in space. –Does not necessarily lie in the physical link. –If axes intersect then it is zero. –Not defined for Prismatic, set to zero z1z1 z2z2 a1a1

31. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. nn Link Twist Angle Measured from z n to z n+1 around x n Usually a multiple of 90 degrees 11 z1z1 z2z2 x1x1

32. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. dndn Link Offset –Measured from x n-1 to x n along z n –Joint variable for Prismatic joint z1z1 z2z2 y1y1 x1x1 y2y2 x2x2 d1d1

33. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. nn Joint Angle The angle between x n-1 and x n Measured around z n from x n-1 and x n z1z1 z2z2 y1y1 x1x1 x2x2  2 x2x2 x1x1

34. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Assigning Coordinate Frames 1.First place the Z axes along the joint axes of rotation, or translation. Imagine them of infinite length 2.Identify the Common Perpendiculars or point of intersection between each pair of Z axes Place the X n axes along the common perpendicular between axes n and n+1. If the axes intersect X n is assigned normal to the plane containing the intersecting axes. 3.Use the right hand rule to define the Y axes. 4.The base frame (0) is fixed and does not move. It is usually picked to be coincident with frame 1 so that when the joint variable is zero: a 0 =0  0 =0Link Variables (Both are constants) d 1 =0  n =0Joint Variables (One is always a constant) 5.Chose the origin and direction of {N} freely, but in general assign it to make as many parameter zero as possible

35. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Special Cases If a n =0 (the Z axes intersect) –then pick X n to be normal to the plane defined by Z n and Z n+1. This leads to two possible Definitions of X and thus Y. –If Z n and Z n+1 are parallel and coincident, then the assignment of X n is arbitrary. If Z n-1 and Z n are parallel then pick d n is 0

36. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. DH Parameters Make a Matrix of the DH parameters for all the links in the robot. This will make defining the transformation from one axes to another much easier.

37. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Forward Kinematics Used to move from Joint coordinates to end-effector coordinates We will use DH parameters to Define Frames –Each Link will be assigned a frame –Transformation will be used to move between links We will start at the bottom and work our way out the chain of links

38. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Transformation between Frames A transformation between Frame {i-1} and {i) can be defined as having 4 steps each containing only one link parameter. 1.translate along z i, by d i 2.rotate around z i, by  i- 3.translate along x i-1, by a i-1, 4.rotate around x i-1, by  i-1 We can perform this with two transforms –First a translation and rotation with regards to z i –Second a translation and rotation with regards to x i-1 Remember the operations will be performed from Left to Right.

39. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Transformations

40. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. DH Based Transformation Matrices

41. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Forward Kinematics Used to mover from Joint coordinates to end-effector coordinates Just Multiply the individual T matrices. For a robot with N joints we will get N Transformations If we are interested in a point beyond the last joint such as an end- effector or tool. We define a coordinate system for the Tool point of interest and create a transform from the last joint to the point. We will find it helpful to define other points in the robot work space with coordinate systems relative to the robot base system.

42. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Standard Frames include Wrist Frame {W} Same as the last robot frame I.e. Frame {N} Base frame {B} Located at the base of the manipulator same as frame {0} Station frame {S} Usually used as the base point for robot motions. Calculated relative to the Base Frame Tool Frame {T} Could be tip of a torch, point between finger of a gripper or the lens of a camera. specified relative to the {W} Frame Goal Frame {G} Location where the robot is desired to move. At the end of motion the {G} and {T} should coincide. So to get from the Station frame to the toll frame we simply calculate: Where

43. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Example Book

44. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Motor Vs Joint May have a gear reduction N is the gear ratio Power is conserved May be nonlinear through a four-bar linkage.

45. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Types of Gear Trains Harmonic Drive –High Gear Ratio –Compact –No Backlash Planetary –Robust –Medium Gear Ration Worm Gear –Right Angle –High Ratio Ball Nut –Rotational to Transnational Ball Screw –Ball nut with lower friction

46. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Harmonic Drive A Harmonic Drive Consists of –Wave Generator –Flex Spline –Circular or Fixed Spline Back Drivable Output Reversing No Backlash As the Lobed Wave Generator Rotates –The Flex Spline is forced into the fixed spline –Since there are more teeth on the fixed spline than the flex, they will rotate relative to each other by the difference in teeth. (Equal to the number or lobes) –Very High ratios 100 to 1 or more can easily be achieved. Pictures From Harmonic Drive Inc.

47. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Ball Screw or Nut Is really a form of a worm gear Has a gear of infinite diameter and a very long worm gear. As the worm rotates the nut moves up the worm (equivilent to rotation) Often described in pitch –Pitch is the distance the nut moves per screw revolution –Units are distance or more precisely mm/rev or in/rev Convert Rotational motion to linear motion. An option to a rack and pinion. High Gear ration. Ball Screws are Back-drivable in Most ratios.

48. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Worm Gear Worm Gears Can Provide a very high Gear Ratio. Each time the input shaft rotates one revolution the gear will move by one tooth. By using a large gear lets say 40 teeth we can get a high Gear ratio 40 to 1. Motor can be mounted at Right Angle to Output shaft Usually not Back-Drivable Picture from www.howstuffworks.com

49. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Planetary The yellow gear (the sun) engages all three red gears (the planets) simultaneously. All three are attached to a plate (the planet carrier), and they engage the inside of the blue gear (the ring). Because there are three red gears, this gear train is extremely rugged. The Output Shaft is usually attached to the blue gear resulting in a 6 to 1 gear ratio. If we used the dark blue planet carrier as the out put we get a 7 to 1 gear ratio since the planets will orbit the sun. If we use the Planets carrier as the input, and the ring as the output and fix the sun, we end up with a 1.17 (7/6) to 1 ratio. Planetary gears are often cascaded to get high gear ratios the blue gear has six times the diameter of the yellow

50. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Assigning Coordinate Frames Review 1.First place the Z axes along the joint axes of rotation, or translation. Imagine them of infinite length 2.Identify the Common Perpendiculars or point of intersection between each pair of Z axes Place the X n axes along the common perpendicular between axes n and n+1. If the axes intersect X n is assigned normal to the plane containing the intersecting axes. 3.Use the right hand rule to define the Y axes. 4.The base frame (0) is fixed and does not move. It is usually picked to be coincident with frame 1 so that when the joint variable is zero: a 0 =0  0 =0Link Variables (Both are constants) d 1 =0  n =0Joint Variables (One is always a constant) 5.Chose the origin and direction of {N} freely, but in general assign it to make as many parameter zero as possible

51. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. Special Cases If a n =0 (the Z axes intersect) then pick X n to be normal to the plane defined by Z n and Z n-1. This leads to two possible Definitions of X and thus Y. If Z n and Z n+1 are parallel then d n is 0

52. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. DH Parameters Make a Matrix of the DH parameters for all the links in the robot. This will make defining the transformation from one axes to another much easier.

53. Feb 17, 2002Robotics 1 Copyright Martin P. Aalund, Ph.D. DH Based Transformation Matrices