A kinematic chain is comprised of a series of linked rigidbodies (bones) where each successive bone is connected to its previous bone by joints, from the root joint to the last joint in the chain; the end effector – as discussed by Kenwright (2013, pp. 2-9).
“The forward kinematics of a manipulator deals with the computation of the position and orientation of the manipulator end-effector in terms of the active joints variables.” (Ezzat A Showaib, 2013, p. 1)
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| Figure 1: Kinematic chain of bones joined by connecting joints (Kucuk and Bingul, 2006, p. 119) |
A kinematic chain is a hierarchical structure. By altering the rotation of any joint, all it’s child joints will be rotated relatively without their individual rotations being altered (as shown in figure 2).
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| Figure 2: Child joint positions are offset relative to the rotation of those higher in the hierarchy (Zucconi, 2017) |
The solution to the problem of forward kinematics for any chain has a single solution, free of ambiguity.
Denavit-Hartenberg Convention
The most prevalent “convention for selecting frames of reference in robotic applications is the Denavit-Hartenberg, or D-H convention” (Spong, Hutchinson and Vidyasaga, 2004, p. 64)This approach attaches reference frames to the joints of a kinematic chain. The relationship between any given joint frame is then compared to the previous joint frame in the chain to describe the location of the joint relative to the previous frame.
Geometric Convention
For kinematic chains of lower degrees of freedom, as Zucconi, A. (2017) notes, the end effector position can be calculated by sequentially summing the bone length and offset from the initial rotation of each sequential joint in the kinematic chain using the following two equations: |
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| Figure 3: End effector rotation (Zucconi, 2017) | Figure 4: End effector position (Zucconi, 2017) |
Application in my Solution
While both methods will return the desired information - for my requirements, relying on the hierarchical structure of Unity will automatically cause the child joints configuration to match that of forward kinematics.As in Unity a direct comparison can be made from the position of the end effector to the target position, there is no need to include any foward kinematic functionality.
This behaviour is shown in figure 5, where the rotation of a joint in the chain effects the global rotation of all lower connected joints while preserving the relative rotations.
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| Figure 5: Unity's hierarchical relationships automatically producing forward kinematics |
References
Ezzat A Showaib, A. (2013). Artificial Neural Network Based Forward Kinematics Solution for Planar Parallel Manipulators Passing through Singular Configuration. Advances in Robotics & Automation, [online] 02(02), pp.1-6. Available at: https://www.omicsonline.org/open-access/artificial-neural-network-based-forward-kinematics-solution-for-planar-parallel-manipulators-passing-through-singular-configuration-2168-9695.1000106.pdf [Accessed 11 Nov. 2018].
Kenwright, B. (2013). Game inverse kinematics. [Place of publication not identified]: CreatSpace, pp. 2-14.
Kucuk, S. and Bingul, Z. (2006). Robot Kinematics: Forward and Inverse Kinematics. Industrial-Robotics-Theory-Modelling-Control, [online] pp.117-148. Available at: https://pdfs.semanticscholar.org/d2ff/f0043238ad6b525ed22839e813e3e85b5511.pdf [Accessed 11 Nov. 2018].
Spong, M., Hutchinson, S. and Vidyasaga, M. (2004). Robot Dynamics and Control. 2nd ed. pp.61-82.
Zucconi, A. (2017). The Mathematics of Forward Kinematics - Alan Zucconi. [online] Alan Zucconi. Available at: https://www.alanzucconi.com/2017/04/06/forward-kinematics/ [Accessed 10 Nov. 2018].
Bibliography
Alamsyah, F. (2019). The Kinematics Analysis of Robotic Arm manipulators Cylindrical Robot RPP Type for FFF 3D Print using Scilab. IOP Conference Series: Materials Science and Engineering, [online] 494, pp.1-7. Available at: https://iopscience.iop.org/article/10.1088/1757-899X/494/1/012100/pdf [Accessed 13 Nov. 2018].
Hock, O. and Šedo, J. (2017). Forward and Inverse Kinematics Using Pseudoinverse and Transposition Method for Robotic Arm DOBOT. Kinematics, [online] pp.75-94. Available at: https://www.intechopen.com/books/kinematics/forward-and-inverse-kinematics-using-pseudoinverse-and-transposition-method-for-robotic-arm-dobot [Accessed 12 Nov. 2018].
