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Stability of Surface Contacts for Humanoid Robots: Closed-Form Formulae of the Contact Wrench Cone for Rectangular Support Areas
Stéphane Caron, Quang-Cuong Pham, Yoshihiko Nakamura
TL;DR
For rigid surface contacts, existing checks of the Contact Wrench Condition rely on many individual contact forces, making computations costly and hindering efficient inverse-dynamics methods. This paper derives closed-form wrench-cone formulae for rectangular supports directly from their geometry. The formulae express friction, ZMP, and yaw-torque conditions, and yield simpler, faster single-support computations demonstrated in OpenHRP.
Problem
Checking the Contact Wrench Condition with many individual contact forces is computationally costly and prevents efficient inverse-dynamics methods.
Source: §1 · p. 2Method
The paper computes the contact wrench cone directly from surface geometry and derives closed-form formulae for rectangular contact areas.
Source: §1 · p. 3Results
The rectangular wrench-cone formulae express Coulomb friction, ZMP, and yaw-torque bounds, and yield simpler, faster computations for humanoid motions in single support demonstrated in OpenHRP.
Source: §4 · p. 8, §6 · p. 11Takeaways & Limitations
The yaw-torque bounds provide an analytical description of yaw friction from which the authors derive a control law to avoid undesired foot rotations.
Source: §1 · p. 3, §6 · p. 11Using a constant friction coefficient is an approximation because the friction coefficient varies across the contact surface.
Source: §3 · p. 6
Abstract
from arXiv · showhide
Humanoid robots locomote by making and breaking contacts with their environment. A crucial problem is therefore to find precise criteria for a given contact to remain stable or to break. For rigid surface contacts, the most general criterion is the Contact Wrench Condition (CWC). To check whether a motion satisfies the CWC, existing approaches take into account a large number of individual contact forces (for instance, one at each vertex of the support polygon), which is computationally costly and prevents the use of efficient inverse-dynamics methods. Here we argue that the CWC can be explicitly computed without reference to individual contact forces, and give closed-form formulae in the case of rectangular surfaces -- which is of practical importance. It turns out that these formulae simply and naturally express three conditions: (i) Coulomb friction on the resultant force, (ii) ZMP inside the support area, and (iii) bounds on the yaw torque. Conditions (i) and (ii) are already known, but condition (iii) is, to the best of our knowledge, novel. It is also of particular interest for biped locomotion, where undesired foot yaw rotations are a known issue. We also show that our formulae yield simpler and faster computations than existing approaches for humanoid motions in single support, and demonstrate their consistency in the OpenHRP simulator.
1 Introduction
ZMP alone is insufficient for contact stability because it does not address sliding or yaw rotation, while force-based criteria account for these but introduce redundant variables. The paper proposes computing the contact wrench cone directly from surface geometry, deriving closed-form formulae for rectangular supports and a yaw-rotation control law.
1 Introduction
- ZMP is necessary but not sufficient for contact stability because it does not address horizontal sliding or yaw rotation.Significant yaw moments during single support can cause undesired foot rotations; upper-body motions have been used to compensate while retaining ZMP.Source: §1 · p. 2
- Force-based stability criteria account for sliding and yaw rotation, but their redundant contact-force variables make equations-of-motion resolution harder.The force vector has three times as many components as contact constraints; this complicates inverse dynamics and has prompted contact approximations or expensive polytope projection.Source: §1 · p. 2
- Computing the wrench cone from contact-surface geometry avoids deriving it from individual contact forces.The contact wrench has minimal dimension six, but prior wrench-cone criteria retained a sufficient-friction assumption that did not account for sliding and yaw.Source: §1 · pp. 2–3
- The closed-form rectangular-support wrench cone yields a yaw-torque bound and supports a control law to prevent undesired foot rotations.Most humanoid feet can be approximated by rectangles.Source: §1 · p. 3
2 Background
The background expresses robot dynamics using individual point-contact forces and explains how contact stability constrains motion and generalized forces. It defines weak stability as the existence of a solution that preserves fixed contact mode.
2.1 Contact Forces and Contact Wrench
- For N point contacts, each three-dimensional contact force enters the robot equations of motion through its translation Jacobian, alongside actuated-joint torques.The dynamics use the inertia matrix, gravity and Coriolis forces, and the joint-selection matrix.Source: §2.1 · p. 3
- With rigid environment and link bodies, their interaction can be represented by a single contact wrench and a six-dimensional contact Jacobian at the link-frame origin.The contact Jacobian stacks the translation and rotation Jacobians.Source: §2.1 · pp. 3–4
2.2 Contact Stability
- Weak contact stability requires a solution to the equations of motion that satisfies fixed contact mode for every contacting link.In fixed contact mode, end-effector position and orientation are equality-constrained to reference values; accelerations and generalized contact forces obey complementary constraints.Source: §2.2 · p. 5
- Inverse-dynamics approaches first enforce conditions on velocity and acceleration kinematically, then solve for torques and complementary forces.The paper describes weak stability as the stability sense used in the remainder of the paper.Source: §2.2 · p. 5
3 Surface Contact
Surface contact is modeled through continuous pressure and tangential-stress fields, whose admissible wrenches can be represented by forces at the vertices of a convex polygon. With constant friction, the paper argues that this vertex representation fully describes surface-contact dynamics, while noting that constant friction approximates spatial variation.
3 Surface Contact
- The paper treats contact forces at points as complementary variables to the contacting link’s position and orientation.It introduces forces f1, . . . , fN for this point-based formulation.Source: §3 · p. 5
- Surface contact is modeled by normal pressure p(x, y) and tangential stress σ(x, y), combined as ν = σ(x, y) + p(x, y)n.The paper uses ν to describe the action of the environment across the contact surface.Source: §3 · p. 6
- Constant μ is an approximation because the friction coefficient varies with position across the surface.Source: §3 · p. 6
- Existing surface-contact models use point sets, sometimes exceeding positional-constraint needs; the proposition theoretically justifies forces at convex-hull vertices.The examples include nearly 30 points per link in and one point at each convex-hull vertex in.Source: §3 · pp. 6–7
- For a convex polygon, any admissible surface-force field has a vertex-force representation with the same contact wrench and complementary constraints.The paper says the full proof that the field and vertex-force conditions are equivalent is outside its scope.Source: §3 · p. 7
- Assuming constant μ makes the field and vertex-force conditions equivalent, and vertex forces then describe surface-contact dynamics.The paper states that the complete proof of equivalence is outside its scope.Source: §3 · p. 7
4 Wrench Cone for Rectangular Surfaces
For a rectangular support area, the contact wrench can be tested directly against a closed-form cone rather than reconstructed from individual vertex forces. Its inequalities combine Coulomb friction, ZMP bounds, and a coupled bound on yaw torque.
4 Wrench Cone for Rectangular Surfaces
- Under the linear friction-cone approximation, Proposition 2 gives an if-and-only-if wrench criterion for forces at the four vertices of a rectangular support area.The wrench is expressed at the rectangle's center in the link reference frame.Source: §4 · p. 8
- The cone's first two inequalities express Coulomb friction, its next three are equivalent to the ZMP condition, and its last bounds admissible yaw torque.The yaw bound is coupled to the other wrench components, rather than requiring only that yaw torque be small.Source: §4 · p. 8
- The admissible yaw torque may deviate from τ_safe only within a bound, and τ_safe is identified as the safest value rather than zero.The supplied passage states the existence of the deviation bound but does not give its expression.Source: §4 · pp. 8–9
5 Experiment
The CWC was implemented in TOPP to generate a single-support motion that challenged all six contact degrees of freedom, then tested in OpenHRP. Retiming approached the contact boundary; acceleration caused loss of surface contact and, at 15%, a fall.
5 Experiment
- The CWC was tested in OpenHRP with an HRP4 model, whose simulator distributes contact forces along surface edges rather than corners.The motion challenged all six contact degrees of freedom in single contact at the left foot.Source: §5 · p. 9
- At 10% acceleration the foot reached point and line-edge contact without falling; at 15%, the robot fell after the CWC violation exceeded the stabilizer’s ability.Figure 4 shows the contact-force distributions; the retimed motion lasted 7.29 s, versus 6.55 s and 6.19 s for the accelerated motions.Source: Figure 4 · p. 10, §5 · p. 11
- The trajectory was built from eleven hand-designed postures, interpolated with Bezier curves, then retimed using the open-source TOPP solver.A kinematic filter fixed the support foot’s ground position and orientation.Source: §5 · p. 10
- TOPP’s time-optimal retiming saturates at least one contact constraint, so in an ideal setting any faster execution would break contact.The experiment showed this boundary behavior, though stabilizer-induced joint-angle changes could under- or over-saturate constraints.Source: §5 · pp. 10–11
- To accommodate tracking deviations, the experiment scaled the contact area by 45% and used friction coefficient µ = 0.4 instead of OpenHRP’s µ = 0.8.With these safety margins, the motion could be accelerated by about 5% and still execute successfully.Source: §5 · p. 11
6 Conclusion
The paper derives a closed-form wrench cone for rectangular contact surfaces and identifies a yaw-friction bound that supports a simple control law. It reports simpler and faster computations than contact-force formulations for humanoid motions in single support, demonstrated in OpenHRP.
6 Conclusion
- The closed-form rectangular-contact wrench cone yields simpler and faster computations than contact-force formulations for humanoid motions in single support, with dynamic motions simulated in OpenHRP.The formula also provides a concise double-inequality description of yaw friction and supports a simple control law to avoid undesired foot rotations.Source: §6 · p. 11
- The yaw-friction double inequality is described as novel and is used to derive a simple control law against undesired foot rotations.The conclusion frames such rotations as a recurring problem for bipeds in single contact.Source: §6 · p. 11
- The paper calculates a closed-form expression of the wrench cone for rectangular contact surfaces.The conclusion presents this as the basis for the computational and yaw-control implications.Source: §6 · p. 11
A Proof of Equation 4
The proof uses the rotation Jacobian to relate a link’s rotation velocity to joint velocities. The supplied passages introduce a further consequence for points on the link but do not include its full expression.
A Proof of Equation 4
- The rotation Jacobian relates link rotation velocity ω to joint velocities ˙q through ω = Jrot ˙q.The passage then introduces a consequence for any point C on the link and any vector u.Source: A Proof of Equation 4 · p. 12
- The passage states that a property holds for any point C on the link and any vector u, but the supplied text does not provide the property itself.The proof context identifies the preceding Jacobian relation as the starting point.Source: A Proof of Equation 4 · p. 12
- The translation Jacobian is introduced after expressing point Ci as the link origin O plus the vector −−→ OCi.The passage gives this positional relation but not the resulting Jacobian expression.Source: A Proof of Equation 4 · p. 12
A.1 Calculation of the Wrench Cone
The calculation reduces the wrench-cone feasibility system by replacing the vertex coefficients α with aggregate variables γ, then eliminates variables to derive bounds on the normalized wrench components.
A.1 Calculation of the Wrench Cone
- The derivation replaces α ∈ [−1, 1]8 with γ in the image of the linear map M, reducing the system while preserving its feasible solutions.The map M is described as a linear surjection from [−1, 1]8 to M[−1, 1]8.Source: §A.1 · p. 13
- The initial reduction yields bounds on γx, γ′x, γy, and γ′y in terms of C1 and C2, with C1 and C2 constrained to [−1, 1].The derivation obtains the latter condition by matching lower and upper bounds.Source: §A.1 · pp. 13–14
- Eliminating the redundant γ variables leaves a feasibility test by matching lower and upper bounds on γx.The text then states that these pairwise comparisons yield the subsequent inequalities.Source: §A.1 · p. 14
- For (K1, K2, C1, C2) ∈ [−1, 1]^4, the final normalized bounds are K3 ≥ −1 + py|K1 − C1| + px|K2 − C2| and K3 ≤ +1 − py|K1 + C1| − px|K2 + C2|.Source: §A.1 · p. 15