所謂的逆運動學(Inverse Kinematics)問題是在尋找一組機械手臂的轉軸角度，使得機械手臂能將末端夾爪以朝向目標方向的姿態呈現於目標位置。一般來說，逆運動學問題能夠透過數值解，以迭代的方式求解。但若目標手臂的自由度數量上升，或是目標任務帶有額外的限制，整個迭代過程所需的時間就會顯著拉長。
本論文提出了一個藉由查找表來來加速數值解型逆運動學計算的方法。透過查找表所提供手臂轉軸角度，手臂的末端點將會被移至鄰近目標的位置，並以該姿勢作為迭代計算的起始姿勢。這樣的作法能夠顯著的減少計算所需的迭代次數以及運算時間。當目標任務有額外的條件時，本論文選擇在進行迭代計算之前就提前滿足那些條件，而非把那些條件加入迭代的計算式中，以減低迭代過程中的運算複雜度。本查找表僅需在機械手臂開發完成時一次性的建立，便能適用於各種不同的數值解逆運動學方法。論文中的實驗也表明，結合查找表的數值解逆運動學方法，其運算速度能夠被顯著的提升。

Inverse kinematics (IK) for a robotic arm solves for the joint variables to move its end effector to a target position with an optional orientation. IK can be solved by iterative numerical methods. However, as the degree-of-freedom (DoF) of the robotic arm increases or the applications impose extra constraints, the iterative process will be lengthened dramatically. In this study, we propose to use a look-up table to accelerate the numerical IK methods. Given a target position, the table outputs a set of joint variables that can move the end effector to the positions near the target, from where the numerical method can start the iterative process. This significantly decreases the number of iterations and the total time in solving IK. In addition, extra constraints of the applications can be checked after table lookup, instead of "formulating into" the numerical formulations. This simplifies the formulations and reduces the computation complexity. The proposed look-up table only needs to be built once when the robotic arm is developed and can be coupled with any numerical IK methods. The experimental results show that the look-up table can significantly accelerate different numerical IK methods.

[1] S. R. Buss, “Introduction to Inverse Kinematics with Jacobian Transpose, Pseudoinverse
and Damped Least Squares methods,” 2009.
[2] B. Siciliano, “Kinematic control of redundant robot manipulators: A tutorial,” Journal of
Intelligent and Robotic Systems 1990 3:3, vol. 3, no. 3, pp. 201–212, 1990.
[3] K. Ayusawa and Y. Nakamura, “Fast inverse kinematics algorithm for large DOF system
with decomposed gradient computation based on recursive formulation of equilibrium,”
IEEE International Conference on Intelligent Robots and Systems, pp. 3447–3452, 2012.
[4] Y. Xia and J. Wang, “A dual neural network for kinematic control of redundant robot ma-
nipulators,” IEEE Transactions on Systems, Man, and Cybernetics, Part B: Cybernetics,
vol. 31, pp. 147–154, 2 2001.
[5] E. Oyama, Nak Young Chong, A. Agah, T. Maeda, and S. Tachi, “Inverse kinemat-
ics learning by modular architecture neural networks with performance prediction net-
works,” Proceedings - IEEE International Conference on Robotics and Automation, vol. 1,
pp. 1006–1012, 2001.
[6] A. D’Souza, S. Vijayakumar, and S. Schaal, “Learning inverse kinematics,” IEEE Inter-
national Conference on Intelligent Robots and Systems, vol. 1, pp. 298–303, 2001.
[7] A. Ramdane-Cherif, B. Daachi, A. Benallegue, and N. Lévy, “Kinematic inversion,” IEEE
International Conference on Intelligent Robots and Systems, vol. 2, pp. 1904–1909, 2002.
[8] Y. Yang and D. Ramanan, “Articulated human detection with flexible mixtures of
parts,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 35, no. 12,
pp. 2878–2890, 2013.
[9] A. Toshev and C. Szegedy, “DeepPose: Human Pose Estimation via Deep Neural Net-
works,” Proceedings of the IEEE Computer Society Conference on Computer Vision and
Pattern Recognition, pp. 1653–1660, 12 2013.
[10] I. Mordatch, K. Lowrey, G. Andrew, Z. Popovic, and E. V. Todorov, “Interactive Control of
Diverse Complex Characters with Neural Networks,” in Advances in Neural Information
Processing Systems (C. Cortes, N. Lawrence, D. Lee, M. Sugiyama, and R. Garnett, eds.),
vol. 28, Curran Associates, Inc., 2015.
[11] D. Holden, J. Saito, and T. Komura, “A deep learning framework for character motion
synthesis and editing,” ACM Transactions on Graphics (TOG), vol. 35, 7 2016.
[12] A. Aristidou, J. Lasenby, Y. Chrysanthou, and A. Shamir, “Inverse Kinematics Techniques
in Computer Graphics: A Survey,” Computer Graphics Forum, vol. 37, pp. 35–58, 9 2018.
[13] D. E. Whitney, “Resolved Motion Rate Control of Manipulators and Human Prostheses,”
IEEE Transactions on Man-Machine Systems, vol. 10, no. 2, pp. 47–53, 1969.
[14] W. A. Wolovich and H. Elliott, “COMPUTATIONAL TECHNIQUE FOR INVERSE
KINEMATICS.,” Proceedings of the IEEE Conference on Decision and Control,
pp. 1359–1363, 1984.
[15] A. Balestrino, G. De Maria, and L. Sciavicco, “Robust Control of Robotic Manipulators,”
IFAC Proceedings Volumes, vol. 17, pp. 2435–2440, 7 1984.
[16] C. W. Wampler, “Manipulator Inverse Kinematic Solutions Based on Vector Formulations
and Damped Least-Squares Methods,” IEEE Transactions on Systems, Man and Cyber-
netics, vol. 16, no. 1, pp. 93–101, 1986.
[17] Y. Nakamura and H. Hanafusa, “Inverse Kinematic Solutions With Singularity Robustness
for Robot Manipulator Control,” Journal of Dynamic Systems, Measurement, and Control,
vol. 108, pp. 163–171, 9 1986.
[18] R. Fletcher, Practical methods of optimization. Wiley, 2000.
[19] D. C. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimiza-
tion,” Mathematical Programming, vol. 45, pp. 503–528, 8 1989.
[20] R. H. Byrd, P. Lu, J. Nocedal, and C. Zhu, “A Limited Memory Algorithm for Bound
Constrained Optimization,” SIAM Journal on Scientific Computing, vol. 16, pp. 1190–
1208, 7 1995.
[21] L. C. T. Wang and C. C. Chen, “A Combined Optimization Method for Solving the Inverse
Kinematics Problem of Mechanical Manipulators,” IEEE Transactions on Robotics and
Automation, vol. 7, no. 4, pp. 489–499, 1991.
[22] B. Kenwright, “Inverse Kinematics – Cyclic Coordinate Descent (CCD),”
http://dx.doi.org/10.1080/2165347X.2013.823362, vol. 16, pp. 177–217, 10 2013.
[23] R. Kulpa and F. Multon, “Fast inverse kinematics and kinetics solver for human-like fig-
ures,” Proceedings of 2005 5th IEEE-RAS International Conference on Humanoid Robots,
vol. 2005, pp. 38–43, 2005.
[24] M. Mahmudi and M. Kallmann, “Feature-Based Locomotion with Inverse Branch Kine-
matics,” Lecture Notes in Computer Science (including subseries Lecture Notes in Ar-
tificial Intelligence and Lecture Notes in Bioinformatics), vol. 7060 LNCS, pp. 39–50,
2011.
[25] A. Aristidou and J. Lasenby, “FABRIK: A fast, iterative solver for the Inverse Kinematics
problem,” Graphical Models, vol. 73, pp. 243–260, 9 2011.
[26] A. Aristidou, Y. Chrysanthou, and J. Lasenby, “Extending FABRIK with model con-
straints,” Computer Animation and Virtual Worlds, vol. 27, pp. 35–57, 1 2016.
[27] M. W. Spong, S. Hutchinson, and M. M. Vidyasagar, Robot modeling and control, 2nd
Edition. 2020.
[28] A. Guttman, “R-trees: A dynamic index structure for spatial searching,” Proceedings of
the ACM SIGMOD International Conference on Management of Data, pp. 47–57, 6 1984.
[29] D. Kraft, A software package for sequential quadratic programming. 1988.
[30] J. Lenareie, “An efficient numerical approach for calculating the inverse kinematics for
robot manipulators,” Robotica, vol. 3, no. 1, pp. 21–26, 1985.
[31] A. A. Maciejewski and C. A. Klein, “Obstacle Avoidance for Kinematically Redun-
dant Manipulators in Dynamically Varying Environments,” The International Journal of
Robotics Research, vol. 4, pp. 109–117, 7 1985