Application In Robotics
In robotics the “Zero moment point (ZMP) is a mathematical formulation to find a point that causes equilibrium of action and reaction momentum (momentum equal to zero) … ZMP can be approached using Single Linear Inverted Pendulum Model (SLIPM).” (Pristovani et al., 2017, p. 1)A SLIPM can therefore be used to determine whether the biped is balanced, and corrective limb placement to reach the ZMP.
To achieve this, the model has two main considerations: the position of the centre of mass and the linear acceleration of the object.
My Solution
As the model in my solution will operate freely from gravity, I do not require the usual functionality that an inverted pendulum yields.Though a greatly simplified version may be able to produce a more natural hip trajectory during locomotion.
In their research, Carlos, Iván and Antonio show the heel as the biped’s natural pivot point during the limbs support phase 2016. As the hips move over the support limb they raise to their apex as the support limb reaches full extension, before descending to their lowest regular value as the swing limb collides with the ground plane.
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| Figure 1: Joint tracking of real human subject walk gait (Carlos, Iván and Antonio, 2016, p. 445) |
Hutter, Siegwartand and Stastny discuss; ‘static’ walking, which they define as a gait featuring repeating double support periods, and how it can be represented by an inverted pendulum (2016). Their visual representation is shown in Figure 2.
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| Figure 2: Inverted Pendulum, 2D static walk (Hutter, Siegwartand and Stastny, 2019) |
This shows that a linear pendulum, which is only able to rotate along the biped’s sagittal plane could be used to replicate the same hip pattern as recorded by Carlos, Iván and Antonio (2016), during the “single support walking phase in which the inverted pendulum moves” (Ackovska and Bozinovski, 2016, p. 4).
To achieve this, I will make use of Unity’s “Transform.RotateAround” function to restrict the angle of rotation to the forward vector of the biped model, then pivot the hips over the contact position of the support limb to create a step.
I expect this behaviour will be straightforward to implement, though will need to maintain the upright orientation of the hips as the spine base is rotated around the support limb end effector.
This solution should remain effective when progressing the solution to 3D interactions, though when walking up stairs the circular hip trajectory which will be given by the “Transform.RotateAround” function may significantly differ from the more elliptical trajectory of a real human gait.
If this occurs, I will need to reserve development time at a later stage to implement an alternate approach.
Next Objective
I am not able to fully create and test this behaviour until limb trajectory (my next goal), the logic checks to define the gait sub-phase order and proportion and the analytical IK solution have been combined.References
Ackovska, N. Bozinovski, S. (2010). BIPED ROBOTS: FROM INVERTED PENDULUM TO PROGRAMMING 12DOF DANCING POSTURES pp.4. Available at http://ciit.finki.ukim.mk/data/papers/7CiiT/7CiiT-01.pdf. [Accessed: 19 Jan. 2019]
Carlos, AG. Iván, MC. Antonio, J. (2016). Analysis of the kinematic variation of human gait under different walking conditions using computer vision, pp.445 - 454. Available at: http://www.scielo.org.mx/pdf/rmib/v38n2/2395-9126-rmib-38-02-00437.pdf. [Accessed: 21 Jan. 2019]
Hutter, M., Siegwart, R. and Stastny, T. (2016). Lecture «Robot Dynamics»: Legged Robots. Available at: https://www.ethz.ch/content/dam/ethz/special-interest/mavt/robotics-n-intelligent-systems/rsl-dam/documents/RobotDynamics2016/7-leggedrobots.pdf. [Accessed: 19 Jan. 2019]
Pristovani, R., Henfri, B., Sanggar, D. and Dadet, P. (2017). Walking strategy model based on zero moment point with single inverted pendulum approach in “T-FLoW” humanoid robot. 2017 2nd International conferences on Information Technology, Information Systems and Electrical Engineering (ICITISEE). [online] Available at: https://ieeexplore-ieee-org.uos.idm.oclc.org/stamp/stamp.jsp?tp=&arnumber=8285498 [Accessed 20 Jan. 2019].
Bibliography
Erbatur, K. and Seven, U. (2007). An inverted pendulum based approach to biped trajectory generation with swing leg dynamics. 2007 7th IEEE-RAS International Conference on Humanoid Robots. [online] Available at: http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.92.4725&rep=rep1&type=pdf [Accessed 19 Jan. 2019].
Kajita, S., Morisawa, M., Miura, K., Nakaoka, S., Harada, K., Kaneko, K., Kanehiro, F. and Yokoi, K. (2010). Biped walking stabilization based on linear inverted pendulum tracking. 2010 IEEE/RSJ International Conference on Intelligent Robots and Systems. [online] Available at: https://users.dimi.uniud.it/~antonio.dangelo/Robotica/2012/helper/bipedWalking_StabilizationLIP.pdf [Accessed 20 Jan. 2019].
Kuo, A. and Donelan, J. (2009). Dynamic Principles of Gait and Their Clinical Implications. Physical Therapy, [online] 90(2), pp.157-174. Available at: https://blog.han.nl/praktijkhuis/files/2013/03/collegecyclus-Ruud-Meulenbroek1704131.pdf [Accessed 19 Jan. 2019].
Technologies, U. (2019). Unity - Scripting API: Transform.RotateAround. [online] Docs.unity3d.com. Available at: https://docs.unity3d.com/ScriptReference/Transform.RotateAround.html [Accessed 19 Jan. 2019].
