Introduction to Robot Kinematics

• The lecture is an introduction to robot kinematics.

PUMA 560 Robot

• This example robot has 6 revolute joints
• Each revolute joint has one degree of freedom (1-DOF) and its movement is defined by angle

Basic Joint Types

• Revolute Joint with 1-DOF, defined by a variable angle
• Prismatic Joint with 1-DOF, defined with linear variables denoted as 'd'
• Spherical Joint uses 3-DOF, defined with variables 𝜃1, 𝜃2, and 𝜃3.

Basic Movements

• Roll: A rotational movement
• Yaw: Sideways movement in a horizontal plane
• Pitch: Up and down movement in a vertical plane

Human Arm

• The human upper limb possesses seven degrees of freedom, excluding the hand
• First Degree: Shoulder Pitch
• Second Degree: Arm Yaw
• Third Degree: Shoulder Roll
• Fourth Degree: Elbow Pitch
• Fifth Degree: Wrist Pitch
• Sixth Degree: Wrist Yaw
• Seventh Degree: Wrist Roll
• The hand distal to the wrist has 23 degrees of freedom
• Each digit (except the thumb) has 4 degrees of freedom for PIP, DIP, and MCP joints, plus abduction/adduction
• Thumb: Due to complex motion, the thumb has 5 degrees of freedom
• Metacarpal Joints: The 4th and 5th digits will have one degree of freedom each

Path Planning

• Forward Kinematics: Determines position given angles
• Requires length of each link and the angle of each joint
• Finds the position of any point using (x, y, z) coordinates
• Inverse Kinematics: Finds angles to achieve a position
• Requires length of each link and the position of a point on the robot
• Finds the angles of each joint needed to obtain that position

Dot Product Quick Math Review

• Geometric representation: A dot B = ||A|| ||B|| cos(θ)
• Matrix representation: A dot B = [ax, ay] dot [bx, by] = ax * bx + ay * by
• A Unit Vector is in the direction of a chosen vector but has a magnitude of 1: uB = B / ||B||

Matrix

• Multiplication: An (m x n) matrix A and an (n x p) matrix B, can be multiplied if the columns of A equals the rows of B
• Multiplication is NOT commutative: AB ≠ BA
• Addition: Add each element to corresponding locations in each matrix

Basic Transformation Overview

• Translation: Moving between coordinate frames for translation along the X-axis

Translation Calculation

• Px = distance between the XY and NO coordinate planes
• VXY = [Vx, Vy], VNO = [VN, VO], P = [Px, 0]
• VXY = [Px + VN, VO] = P + VNO

Basis Vectors

• Basis vectors are unit vectors that point along a coordinate axis
• n = unit vector along the N-axis
• o = unit vector along the O-axis
• |VNO| = Magnitude of the VNO vector
• VNO = [VN, VO] = [|VNO|cosθ, |VNO|sinθ] = [|VNO|cosθ, |VNO|cos(90-θ)] = [VNO · n, VNO · o]

Rotation

• Rotation (around the Z-Axis)
• θ = Angle of rotation between the XY and NO coordinate axis
• VXY = [VX, VY] and VNO = [VN, VO]
• VX = |VXY| cos(α) = |VNO| cos(α) = VNO · x
• VX = (VN * n + VO * o) · x
• VX = VN(x · n) + VO(x · o) = VN(cosθ) + VO(cos(θ + 90)) = VN(cosθ) - VO(sinθ)
• VY = VNO sin(α) = VNO cos(90 - α) = VNO · y
• VY = (VN * n + VO * o) · y
• VY = VN(y · n) + VO(y · o) = VN(cos(90 - θ)) + VO(cosθ) = VN(sinθ) + VO(cosθ)

Rotation Matrix

• Written in Matrix Form after rotation:
  • VX = VN(cosθ) – VO(sinθ)
  • VY = VN(sinθ) + VO(cosθ)
• The Rotation Matrix about the z-axis VXY = [VX, VY] = [[cosθ, -sinθ], [sinθ, cosθ]] [VN, VO]

Combining Rotation and Translation

• VXY = [VX, VY] = [Px, Py] + [[cosθ, -sinθ], [sinθ, cosθ]] [VN, VO]
• A translation followed by a rotation is different from a rotation followed by a translation
• With the coordinates of a point (VN, VO) in some coordinate frame (NO), the position of that point can be found relative to the original coordinate frame (X0Y0)

Homogeneous Representation

• Putting into a Matrix
• VXY = [VX, VY] = [Px, Py] + [[cosθ, -sinθ], [sinθ, cosθ]] [VN, VO]
• Padding with 0s and 1s:
  • [VX, VY, 1] = [Px, Py, 1] + [[cosθ, -sinθ, 0], [sinθ, cosθ, 0], [0, 0, 1]] [VN, VO, 1]
• Simplifying:
  • [VX, VY, 1] = [[cosθ, -sinθ, Px], [sinθ, cosθ, Py], [0, 0, 1]] [VN, VO, 1]
• Homogenous Matrix: The XY plane is translated and then rotated around the z-axis
  • H = [[cosθ, -sinθ, Px], [sinθ, cosθ, Py], [0, 0, 1]]

Rotation Matrices in 3D

• Rotation around the Z-Axis
  • Rx = [[cosθ, -sinθ, 0], [sinθ, cosθ, 0], [0, 0, 1]]
• Rotation around the Y-Axis
  • Ry = [[cosθ, 0, sinθ], [0, 1, 0], [-sinθ, 0, cosθ]]
• Rotation around the X-Axis
  • Rz = [[1, 0, 0], [0, cosθ, -sinθ], [0, sinθ, cosθ]]

Homogenous Matrices in 3D Overview

• A 4x4 matrix that describes a translation, rotation, or both:
  • Translation without rotation: H = [[1, 0, 0, Px], [0, 1, 0, Py], [0, 0, 1, Pz], [0, 0, 0, 1]]
  • Rotation without translation: H = [[nx, ox, ax, 0], [ny, oy, ay, 0], [nz, oz, az, 0], [0, 0, 0, 1]]
  • Rotation can be around z-axis, y-axis, x-axis, or a combination
  • The (n,o,a) position of a point is relative to the coordinate frame
• If and only if both the rotation and translation are relative to the same coordinate frame, a single homogeneous matrix can get created when combining rotation and translation

Finding the Homogeneous Matrix

• [WX, WY, WZ] = [WI, WJ, WK] where W is point relative to the X-Y-Z frame and I-J-K frame, respectively
• [WI, WJ, WK] = [Pi, Pj, Pk] + [[ni, oi, ai], [nj, oj, aj], [nk, ok, ak]] [WN, WO, WA]
• [[WI, WJ, WK], 1] = [[ni, oi, ai, Pi], [nj, oj, aj, Pj], [nk, ok, ak, Pk][0, 0, 0, 1]] + [[WN, WO, WA], 1]
• (Translation relative to the XYZ frame) * (Rotation relative to the XYZ frame) * (Translation relative to the IJK frame) * (Rotation relative to the IJK frame)

Finding Homogenous Variation

• H = (Rotate so that the X-axis is aligned with T) * (Translate along the new t-axis by || T || (magnitude of T)) * (Rotate so that the t-axis is aligned with P) * (Translate along the p-axis by || P || ) * (Rotate so that the p-axis is aligned with the O-axis)
• This can be an easier method to use

The Situation for Forward Kinematics

• A robotic arm that starts aligned with the xo-axis is moved by telling the first link to move by θ1 and the second link to move by θ2
• To find the position of the end of the robotic arm

Forward Kinematics Solutions

• Geometric Approach, easiest but gets tedious for complex geometries
• Algebraic Approach, involving coordinate transformations

Forward Kinematic Example

• A three-link arm starts aligned to the x-axis where each link has lengths l1, l2, and l3
• The homogeneous matrix to express the position of the yellow dot in the X0Y0 frame can be found with:
  • H = Rz(θ1) * Tx1(l1) * Rz(θ2) * Tx2(l2) * Rz(θ3)
  • i.e. rotating by 01 puts you in the X1Y1 frame, translating along the X1 axis by l1, and rotating by 02 puts you in the X2Y2 frame until you are in the X3Y3 frame
• To find the position of the yellow dot relative to the X0Y0 frame, multiply H by the position vector (l3, 0), the position of the yellow dot relative to the X3Y3 frame.

Variation

• Make the yellow dot the origin of a new coordinate X4Y4 frame
• H = Rz(θ1) * Tx1(l1) * Rz(θ2) * Tx2(l2) * Rz(θ3) * Tx3(l3) takes you from the X0Y0 frame to the X4Y4 frame where the position of the yellow dot relative is (0,0)
• Multiplying by the (0,0,0,1) vector will equal the last column of the H matrix.

Denavit-Hartenberg (D-H) Parameters

• Each joint is assigned a coordinate frame
• 4 parameters needed to describe how a frame (i) relates to a previous frame ( i -1 ):
  • α: Angle
  • a: a
  • d
  • 𝜃: Theta

D-H Technical Definitions

1. a(i-1) ​:
   • The length of the perpendicular between the joint axes
   • The joint axes are the axes around which revolution takes place, Z(i-1) and Z(i)
   • The common perpendicular is the shortest line between the two axis-lines and is perpendicular to both axis-lines
2. α(i-1):
   • The amount of rotation around the common perpendicular
   • It rotates around the X(i-1) axis to have the Z(t-1) pointing in the same direction as the Zi axis
   • Follow the right-hand rule for positive rotation
3. d(i-1):
   • The displacement along the Zi axis needed to align the a(i-1) common perpendicular to the ai common perpendicular
   • It aligns the X(i-1) and Xi axes-lines
4. θi​:
   • The amount of rotation around the Zi axis needed to align the X(i-1) axis with the Xi axis

D-H Visual Definitions

• Visual Approach: The link parameter a(i-1) imagines an expanding cylinder whose axis is the Z(i-1) axis
• The radius of the cylinder is equal to a(i-1) when the cylinder touches joint axis i
• Diagram: The common perpendicular is usually the X(i-1) axis
• The displacement along the X(i-1) is how you would move from the (i-1) frame to the iframe
• If the link is prismatic, then a(i-1) is a variable, not a parameter

D-H Matrix

• The Denavit-Hartenberg Matrix is a transformation matrix from one coordinate frame to the next
• A series of D-H Matrix multiplications and the D-H Parameter table gives the final result, a transformation matrix from some frame to your initial frame
• Matrix form: ![[Pasted image 20240226183931.png]]

D-H Example Setup

• A table describes the robot with variables and parameters, describing some state of the robot when having numerical values for the variables
• 3 Revolute Joints can be expressed with:
  • The translation of OTo1, OTo1, and 1To2
  • T=(0T)(1T)(2T) where T is the D-H matrix with (i-1) = 0 and i =1

D-H Example

• General form ![[Pasted image 20240226184053.png]]
• Specific example ![[Pasted image 20240226184113.png]]
• T=(0T)(1T)(2T)

Inverse Kinematics - finding angles from position

• Revolute and Prismatic Joints Combined
• 0 = arctan(y/x)
• S = √(x^2 + y^2)

Two Link Manipulator

• Given l1, l2 x,y find 01, 02
• A unique solution may not exist and multiple may be possible or no solution may be possible.

Two Link Manipulator - Using the Law of Cosines

![[inverse kinematics of a 2 link arm with geometric solution]] ![[]]

Two Link Manipulator - Using The Algebraic Solution

![[inverse kinematics of a 2 link arm with algebraic solution]]

Matlab Demo

• Will have examples

Joints Problem

• How may joints are in the following Robots
• what type of joints they are

Solution using DH parameters

• Describe Forward Kinematics using DH parameters*