Contact Wrench Cone
Continued reading into approaches to maintain biped stability revealed some limitations of ZMP concept.As discussed by Dai, H. and Tedrake, R. (Planning robust walking motion on uneven terrain via convex optimization, 2016) The ZMP approach relies on the assumption that the ground contact plane is perfectly flat and even, otherwise the calculation of the support polygon can quickly become highly complex and produces poorly defined areas.
Dai, H. and Tedrake, R. continue to explain that ZMP always considers the contact surface to have theoretically infinite friction to keep the leg limbs from slipping on the surface. This is an issue in real-world robotics as it means that even if the ZMP lies within the support polygon, the leg limb can still slide along the surface and unbalance the system.
In real-world robotics, slipping is a necessary consideration. For my simulation I would be able to adjust the friction of any surface to be sufficient to guarantee no leg-slip, though expanding the solution to assess friction would allow for more believable interactions with external objects as the previous proposed blog improvement would allow.
The stability of the biped systems stance on surfaces of varying friction and tilt can be assessed with the concept of a contact wrench cone.
“The contact wrench cone (CWC) is the admissible set of the total contact wrench, which is computed by summing up the individual contact wrenches at each contact location.” (Dai, H. and Tedrake, R., 2016, P. 580), where a wrench is the concatenation of force and torque.
Figure 1 shows the construction of a CWC. Friction wrenches from each foot contact point are combined to give a convex hull: the contact wrench cone.
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| Figure 1: The linearized friction cones at each corner of the feet (Dai, H. and Tedrake, R., 2016, P. 580) |
Contact Wrench Cone Margin
The “Contact Wrench Cone margin is the smallest magnitude of the wrench disturbance being applied at a certain location, that the robot cannot resist, given the contact locations and friction cone constraints.” (Dai, H. and Tedrake, R., 2016, P. 581).Geometrically the CWC margin is the distance from the position of the contact wrench of the biped to the CWC boundary.
So long as the position of the contact wrench of the biped lies within the CWC boundaries of its ground contact points the biped will not slip and lose balance.
As the CWC boundary is approached, to avoid falling into limb-slip the CWC margin can be increased by minimising “the centroidal angular momentum in order to obtain a natural walking motion” (Dai, H. and Tedrake, R., 2016, P. 583) with linear constraints. This cannot be done directly and so the upper bound of the centre of mass is minimised to achieve the same outcome.
As is demonstrated in figure 2, on tilted ground a CWC is able to adapt the biped motion to maintain balance within the permissable range as allowed by the CWC margin, whereas on the same terrain a ZMP will exceed the CWC boundary, slipping and causing the biped will fall.
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| Figure 2: Comparison of ZMP concept and CWC concept on uneven terrain (Dai, H. and Tedrake, R., 2016, P. 584) |
References
Dai, H. and Tedrake, R. (2016). Planning robust walking motion on uneven terrain via convex optimization. 2016 IEEE-RAS 16th International Conference on Humanoid Robots (Humanoids), [online] pp.579-586. Available at: http://groups.csail.mit.edu/robotics-center/public_papers/Dai16a.pdf [Accessed 20 Jul. 2019].
Planning robust walking motion on uneven terrain via convex optimization. (2016). [video] Directed by H. Dai and R. Russ. Available at: https://www.youtube.com/watch?v=4HUVV2gYMb8 [Accessed 20 Jul. 2019].
Bibliography
Modern Robotics, Chapter 12.2.1: Friction. (2018). [video] Directed by K. Lynch. Cambridge University: Northwestern Robotics. Available at: https://www.youtube.com/watch?v=FY92f7gssWc [Accessed 20 Jul. 2019].
