Forces acting on a biped robot. Center of pressure-zero moment point
Summary (3 min read)
Introduction
- In addition, this quite useful concept has not been completely explored, and unfortunately some misinterpretations are sometimes encountered in the literature.
- The authors refer the point associated with contact forces as CoP, while ZMP is considered to be related to gravity plus inertia forces.
- Strict definitions of CoP and ZMP are specified in Section II, and concise algebraic relationships for the computation of both points are formulated.
- The concepts of CoP and ZMP are quite useful for the control of the dynamic equilibrium of bipeds, but first the exact meaning the authors attach to these notions has to be defined.
A. Definitions
- The forces acting on a walker can be separated in two categories: 1) forces exerted by contact and 2) forces transmitted without contact (gravity and, by extension, inertia forces).
- As on the other hand, the resultant of the pressure forces is directed along , then one axis exists, where the moment vanishes at every point of this axis.
- Therefore, the CoP can also be defined as the point on the sole where the moment of the contact forces is perpendicular to the sole (6) (7) where the superscript denotes contact force and moment.
- One can remark that is a noncentral axis of the contact force wrench.
- The ZMP is the point on the ground where the tipping moment acting on the biped, due to gravity and inertia forces, equals zero, the tipping moment being defined as the component of the moment that is tangential to the supporting surface.
B. Expressions of the CoP and ZMP
- The CoP and the ZMP, as defined above, can be computed as follows.
- Knowing the expression of the pressure forces about the point , in other words knowing , the problem consists in determining the position of the CoP.
- Therefore, (11) and consequently the vector can be expressed as (12) Because of the opposition between the gravity-inertia forces and the contact forces, the ZMP is defined by an expression similar to (13) (20).
- The widely favored formulations often met in literature, which one can sum up as (21) are only true if the ground is horizontal, i.e., if with .
A. Demonstration of the Coincidence
- Because of (18) and (19), it is obvious that the axes and coincide (indeed, they are noncentral axes defined from two opposite wrenches).
- The system becomes under-actuated.
- The definitions are unfounded, for the double support phase, if the biped feet are contacting two noncoincident surfaces of an uneven terrain.
- B. Interpretation of the Coincidence With regard of their definitions, the coincidence of CoP and ZMP is not surprising since they are two interpretations of acting force-moment between the ground and the first link of a kinematic chain.
- So, to conclude this section, in order to delete the misinterpretations appearing in literature and subsisting in some minds, the authors can say that, as long as all the ground-sole contacts appear in a single plane surface, then the CoP and the ZMP are absolutely and definitely the same point, that consequently they call CoP-ZMP.
C. Control Aspects
- The unilaterality of the foot-ground contact is a major constraint of legged locomotion.
- Of course, when the biped is flying, the support polygon disappears, and consequently the CoP-ZMP is not defined.
- The major advantage of the CoP-ZMP concept is that this point can be measured: measuring the contact pressure forcemoment allows the CoP to be reconstructed, and the ZMP by coincidence, and therefore the corresponding part of the gravityinertia forces.
- Indeed, the authors use two types of control, the first being assumed acting on the CoP (denoted as C-ATGR in their paper) by “lowering” the heel or the toe of the foot, the second being assumed acting on the ZMP by increasing the magnitude of the inertia forces (accelerating the trunk position).
- What the authors do not say is that a modification of the CoP causes instantaneously a ZMP modification, and vice versa, because the two points coincide.
IN THE CASE OF UNEVEN TERRAIN
- The CoP and ZMP concepts use in their fundamental definitions the vector normal to the ground surface (for the definition of axes and ), and the ground plane itself (which intersects the axes).
- The Honda US Patent [10] tackles the matter of irregular terrain by defining a virtual ZMP and a virtual surface varying continuously from the first to the second surface during the weight transfer from one foot to the other (see Fig. 3).
- The authors have chosen a function proportional to the duration of the double support phase, .
- The virtual surface is defined by the ZMP and by the normal vector , resulting of an identical weighting function as that of , such that (24) Indeed, one can show that if the point is defined by (23), then the contact moment is not directed along defined by (24).
B. Proposition Respecting the CoP-ZMP Concept
- The authors suggest to define a virtual surface equivalent to the two real surfaces and , then to chose the pseudo-CoP-ZMP lying in this surface such that the moment of the contact forces is perpendicular to .
- One must notice that, whatever the weight factors and are, one neither gets (32) nor (33).
- A contrario, the method the authors propose to define the virtual surface and the pseudo-ZMP is based on the definition of the CoPZMP.
- Moreover, the weight factors the authors suggest in (26) give good results, in accord with (32) and (33).
- A contrario, if the two planes are very angled, then it is in their mind necessary to monitor the whole contact forces, as pointed out in Section III-C (see [15]).
C. Case of Noncoincident Parallel Surfaces (Stairs)
- In stairs, the feet are supported by parallel surfaces that have different elevations.
- During the double support phase, the virtual surface is naturally parallel to the others.
- The pseudo-CoP-ZMP is a weighting function of the local CoP-ZMP’s and , such that where (34).
- Whatever the weight factors are, the CoP-ZMP concept (moment of the contact forces perpendicular to the surface), is respected.
- From analogy with the previous case, a good choice is to take the weight factors proportional to the local pressure forces and , such as (35).
V. CONCLUSION AND PERSPECTIVE
- The formal study presented in this paper has established strict definitions about the concepts of CoP and ZMP, clarifying certain misinterpretations sometimes encountered in the literature, and has proposed the concept of pseudo-CoP-ZMP related to walking on uneven terrain.
- As a perspective of this theoretical study, from a practical point of view, one must note that, among the numerous biped robots built world-wide, few, if any, are provided with anthropomorphic soles (the same is true in the case of Bip, their biped robot).
- Experimental results answering to these questions are presented in the companion paper [5], as well as results linked to walking on uneven terrain and on stairs, showing the evolution of the pseudo-CoP-ZMP defined in this paper.
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Citations
343 citations
Cites methods from "Forces acting on a biped robot. Cen..."
...The flat surface approach was first proposed by Takanishi et al. (1990) and later used by Sardain and Bessonnet (2004) ....
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285 citations
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Cites background from "Forces acting on a biped robot. Cen..."
...The “projection of the center of mass” criterion cannot therefore discriminate correctly cases where the system can remain static from cases where it can’t....
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171 citations
References
2,050 citations
"Forces acting on a biped robot. Cen..." refers background in this paper
...Of the earlier works related to control carried out with the ZMP concept, only one is referred to here: in the Honda biped robots, an application of the CoP-ZMP control has been implemented, showing that the CoP notion is related to contact forces, and that of the ZMP to gravity plus inertia forces [9]....
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...It should be noted that this particular axis does not pass through the global CoM , although some authors draw as such in their diagrams (as in [8] and [9] ), and although some stability criteria are based on the position of the projection of the CoM along (see [13] for a bibliography)....
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...It is a key point in the control of the Honda biped robots [9]....
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625 citations
"Forces acting on a biped robot. Cen..." refers background in this paper
...The ZMP concept was introduced and developed in [6] and [7]....
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559 citations
546 citations
"Forces acting on a biped robot. Cen..." refers background in this paper
...The ZMP concept was introduced and developed in [6] and [7]....
[...]
528 citations
"Forces acting on a biped robot. Cen..." refers background in this paper
...For a complete bibliography, see [8] and [13]....
[...]
...It should be noted that this particular axis does not pass through the global CoM , although some authors draw as such in their diagrams (as in [8] and [9] ), and although some stability criteria are based on the position of the projection of the CoM along (see [13] for a bibliography)....
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...It has been exhaustively reviewed in [8]....
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Related Papers (5)
Frequently Asked Questions (6)
Q2. What is the field of pressure forces normal to the sole?
The field of pressure forces (normal to the sole) is equivalent to a single resultant force, exerted at the point where the resultant moment is zero.
Q3. What are the forces acting on a walker?
The forces acting on a walker can be separated in two categories: 1) forces exerted by contact and 2) forces transmitted without contact (gravity and, by extension, inertia forces).
Q4. What is the advantage of the CoP-ZMP concept?
The major advantage of the CoP-ZMP concept is that this point can be measured: measuring the contact pressure forcemoment allows the CoP to be reconstructed, and the ZMP by coincidence, and therefore the corresponding part of the gravityinertia forces.
Q5. What is the resultant of the gravity plus friction forces?
The resultant of the gravity plus inertia forces (superscript ) may be expressed as(8)and the moment about any point as(9)where is the total mass, is the acceleration of the gravity, is the center of mass (CoM) of the biped, is the acceleration of , and is the rate of angular momentum at .
Q6. What is the difference between the virtual and the cop?
According to the authors, the virtual ZMP is a weighting function of the local ZMP’s and , such that(23)where is a function varying continuously from 0 to 1.