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Lecture 3 - Forward Kinematics

What is Forward Kinematics?

Determining the position and orientation (pose) of the end-effector with respect to the base as a function of each joint motion (q which is either θ or d)

Kinematic Chains

j(i)l(i)l(i-1)l(i) moves by q(i) when j(i) is actuatedj(i) fixed to l(i-1)j(1)l(0)j(n)l(n)o(i)x(i)y(i)z(i)o(n)x(n)y(n)z(n)o(i-1)x(i-1)y(i-1)z(i-1)o(0)x(0)y(0)z(0)joint is fixed at the end of each link and moves the next linkframe is fixed at any point along the length of the link (it can be put off the link)first link is the base link and last link is the manipulatoro(i) = o(i-1); how do I go to the next frame (i) from my current frame (i-1)?same index (0 to n)same index (1 to n) asq(i)q(n)q(1)

Transformations

Going to the next frame (i) from the previous frame (i−1)

Ai=[Rii−1dii−101]

Going to the end-effector frame (n) from the base frame (0)

Hn0=∏i=1i=nAi(qi)

Forward Kinematics Approaches

Geometric Approach

Pasted image 20260311061432.png

DH Convention

Frames

  1. Assign axes on the joints
  2. Assign axes on the last important link (manipulator)

4. assign y using right hand rule1. assign z axes according to jointrevolute -> axis of rotationprismatic -> axis of translation2. assign x axis of the base in any direction3. assign the x axis of the next link such that:it is perpendicular to the current z axis (obviously) AND the previous z axisit intersects the current z axis (obviously) AND the previous z axis5. assign the axes of the last important link such that:z axis is parallel to the z axis of the previous linkx axis follows the same rule as mentioned in 3

Transformation Parameters

We are just moving from the origin of the current frame to the origin of the next frame

It's like asking: how can we make the current frame become in the same orientation and position as the next frame?

We just do it in 4 steps

Motion Parameter Name
Rotating around the current z axis to make the x axis of the current frame parallel to the x axis of the next frame θ Joint angle
Translating along the current z axis to intersect with the x axis of the next frame d Joint offset
Translating along the new x axis (x axis of the next frame) to arrive at the origin of the next frame a Link length
Rotating around the new x axis of the next frame to make the current z axis align with the next z axis α Link twist
Tii−1=TRot,zi−1,θi TTran,zi−1,di TTran,xi,ai TRot,xi,αi

This results in a total homogenous transformation matrix from frame i−1 to frame i

To get the transformation from the base frame to the end effector frame we just multiply the transformations between each two frames together

Tn0=T10 T21 T32…Tn−2n−3 Tn−1n−2 Tnn−1

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