Ambiguous collision outcomes and sliding with infinite friction in models of legged systems

1. Introduction

The feet of legged systems have to alternate between aerial phases in which they are moving freely through the air and contact phases in which they are exerting forces against the ground. The focus of this paper is the dynamics of the touchdown process between these two phases. During touchdown, a collision rapidly brings the velocity of the contacting foot to a stop. This collision is the result of a complex visco-elastic interaction between the feet and the ground.

Since it is infeasible to model this interaction in all its physical detail, rigid body dynamic simulations approximate contact either through a simplified spring-damper model or by assuming that the contact is completely rigid (Bicchi and Kumar, 2000). Both approaches are only an approximation of the true underlying continuum mechanics process and provide a simplified means to calculate contact forces and contact accelerations. In spring-damper models, contact forces arise automatically as soon as a foot starts to penetrate into the ground and vanish again once the foot lifts off. This removes the need to keep track of the contact state, but the large stiffness values and the high damping coefficients that are required in such a spring-damper model can be difficult to tune and yield numerically stiff differential equations (Marhefka and Orin, 1999). For simplicity and performance, simulations of legged robots thus often employ completely rigid contact models. These rigid contact models rely on a set of unilateral contact constraints through which vertical foot positions are limited to positive values. If a foot is in contact, the associated contact forces are computed analytically such that the foot does not penetrate into the ground. In this case, the legged system is described as a non-smooth dynamical system, in which a set of unilateral constraints is imposed on the state of the system (Brogliato, 1999). The advantage of such a formulation is that no stiffness parameters must be tuned. This property is particularly useful for template models of locomotion, in which model simplicity and a small number of model parameters are imperative (Garcia et al., 1998). One way to deal with the non-smooth dynamics is to implement a hybrid dynamic simulation framework (Johnson et al., 2016; Goebel et al., 2009; Remy et al., 2011) that explicitly determines lift-off and touchdown events. Furthermore, when a touchdown is detected, the simulation must be interrupted and a set of impulsive forces is computed such that the contacting foot is instantaneously brought to a halt.

In this work, we investigate this process specifically for models of legged robotic systems. Within this context, our primary contribution is to survey the topic of contact ambiguity and to introduce the associated effects to the legged robotics community. In particular, we show how rigid unilateral constraints can lead to a number of non-intuitive behaviors in these systems. Our work highlights the fact that the widely used assumption of no sliding of the contact point can yield mathematically inconsistent solutions when combined with a rigid contact model. We show that, even if an infinite coefficient of friction is assumed, sliding must be included as a potential collision outcome. Furthermore, we identify a range of contact configurations for legged systems in which the rigid body dynamic model produces ambiguous results. Multiple valid solutions of the collision process exist and it is impossible to determine uniquely which state the model will enter after the collision. By investigating these effects, the paper highlights the need for a careful simulation of the collision processes which must go beyond the simple assumption of no sliding. The paper builds on a tutorial for a summer school held at ETH Zurich (Hutter, 2011) in which the basic phenomenon of sliding with infinite friction was reported for a minimalistic model.

The effects described in this paper are not uncommon in rigid body systems that are subject to unilateral constraints. The most famous example is the Painlevé paradox, which has been studied extensively since 1895 (Génot and Brogliato, 1999; Painlevé, 1895). Painlevé’s problem describes a simple rod that is in sliding contact with a moving surface. If the angle between rod and surface is large enough, one is unable to find a set_ of contact forces that prevent the rod from accelerating into the surface. Painlevé’s paradox can only be overcome if impulsive forces and discontinuous velocities are explicitly allowed in this otherwise continuous dynamic system (Stewart, 1997). The behavior is caused by non-zero off-diagonal terms in the contact space mass matrix (see Ruina, 2014, for a great explanation). Such off-diagonal terms can be present anytime when forces are not directly applied to the center of gravity of a body or of a system of bodies. As a consequence of these off-diagonal terms, contact point accelerations and related forces are not generally collinear. This means, that a contact force that is directed away from the contact surface can actually make the contact point move towards the surface and vice-versa. Examples of related phenomena showed the possibility of sliding with infinite friction (as for example in the manipulation of rigid objects Lynch and Mason, 1995) or ambiguous accelerations in multi-contact situations Lötstedt, 1981; Wang et al., 1992. To simulate systems in which these and other phenomena can arise, great care must be taken (Brogliato, 1999, Chapter 5.7) and a number of specialized algorithms have been developed to cope with these phenomena (Anitescu and Potra, 1997; Glocker, 1995; Stewart, 2000; Stewart and Trinkle, 1996).

After introducing the model and basic collision process (Section 2), the paper explores the collision outcomes of a minimalistic bipedal model (Section 3). We provide explanations for the non-intuitive behaviors (Section 4) and show the same effects in the models of a passive dynamic walker (Section 5) and of the bipedal robot RAMone (Section 6).

2. Collision modeling

This paper investigates the collision process in three different bipedal systems (Figure 1). The first is a minimalistic model, in which the mass of each leg is concentrated at a single point located halfway between the hip and the contact point. This model serves to illustrate the dynamics of the collision process in their most basic form. The second system is a passive dynamic walker (PDW) in which the legs have extended mass and a variable center of gravity (COG) location. It is used to highlight the influence of system parameters onto the outcome of the collision process. Finally, we investigate a planar model of the robot RAMone (Smit et al., 2017), a typical planar five-link biped, to show that similar effects arise in actual legged robots.

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Fig. 1.

This paper studies the collision dynamics of three bipedal systems: a minimalistic model, in which the mass of each leg is concentrated in a single point (a) a passive dynamic walker (b) and a model of the bipedal robot RAMone (c). The dynamics of these models are examined as the leading foot (‘le ’) is about to strike the ground in a contact collision. Note that all three bipeds are modeled as moving towards the left in order to yield a positive angular velocity φ . tr of the trailing leg.

Using a floating base description, the configuration of each of these models is given by a vector of generalized coordinates q . The positions of the contact points of the trailing and leading feet are c tr = x tr , y tr T and c le = x le , y le T , respectively. These positions are a function of q ( c = c tr , c le T = c q ) and their velocities are given by

with the contact Jacobian J . The touchdown collision happens in an infinitesimally short amount of time. To compute its outcome, the equations of motion are integrated over this instant (Glocker, 2001)

The superscripts ‘−’ and ‘+’ denote values right before (pre-impact) and right after (post-impact) the instant of the collision. In the limit of an infinitesimally short collision, the integral of the centrifugal, Coriolis, and gravitational forces h disappears. Themass matrix M , then relates changes in velocities to contact impulses Λ . These impulses are mapped from the contact space into the generalized coordinates with the transpose of the contact Jacobian J . By multiplying this equation with J ⋅ M − 1 , the impact is projected into the contact space

or simpler

with the contact space mass matrix M λ

For simplicity, we partition this equation into terms for the trailing foot tr, leading foot le, and coupling terms cp

Analytic expressions for the contact space mass matrix M λ and the pre-impact contact point velocities c . − are derived for the minimalistic model and the PDW in Appendix B.1. At the instance of the collision ( y le = 0 ), the leg angles for these two models are equal and opposite: φ le = − φ tr . We thus simplify our notation to φ tr = φ and φ le = − φ , using φ as a shorthand for the leg angle at impact. This angle is the main parameter that is studied in this paper and it will be varied between 0 and 1 2 π as we evaluate collision outcomes.

3. Collision outcomes

To compute the outcome of the collision, we need to determine the impulses Λ and the post-impact velocities c . + in equation (6). To do so, additional assumptions must be made about the nature of the contact. While there are theoretical limitations to the values that these two quantities can take (see, e.g. Chatterjee and Ruina, 1998), we ultimately have to rely on empirical models that provide a reasonable approximation of the underlying complex continuum mechanics processes and that can be used to relate impulses and post-impact velocities in a physically meaningful way.

In the world of robotics, it is thus often assumed, that the collision is fully plastic with a coefficient of restitution of zero. The choice of the restitution law (e.g. Newton’s law or Poisson’s hypothesis) is thus irrelevant. Furthermore, the majority of simulation studies of legged robots, and in particular of PDWs, assume an infinite coefficient of friction of the ground contact (e.g. Chevallereau et al., 2005; Garcia et al., 2000; Garcia, 1999; Kuo, 1999; McGeer, 1990a,b; Remy et al., 2011, 2009; Saglam and Byl, 2013; Westervelt et al., 2003; Wisse et al., 2004).

3.1. No sliding

Intuitively, these two assumptions suggest that the colliding feet come to a complete stop (or remain at rest) after a perfectly inelastic collision. No sliding is considered. To keep things simple, we will look initially only at cases in which the leading foot comes to a complete stop after the collision. That is, c . le + = 0 . We will revisit this assumption and discuss its validity at the end of this section. This reduces the collision computation to finding the values of Λ tr , Λ le and c . tr + in equation (6). Under the assumption of no sliding, two collision scenarios are possible (Figure 2).

Case 1. As the leading foot hits the ground in the collision, the trailing foot comes off the ground. This outcome is expected to happen when the feet are closer together and the momentum of the walker is pointed forward rather than downward. In this case, impulses at the trailing foot are zero Λ tr = 0 and equation (6) simplifies to c . tr + = M tr − 1 M cp c . le − .

Case 2. After the collision, the trailing foot will remain in place and the system will come to a complete stop. As in this case c . tr + = 0 , we compute the trailing foot impulse as Λ tr = − M cp c . le − . This outcome is expected to happen when the feet are further apart and the momentum of the walker is pointed downward rather than forward.

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Fig. 2.

Under the assumption of no sliding, two collision scenarios are possible. The trailing foot could either leave the ground (case 1) or remain on the ground (case 2). In the first case, no impulse is applied at the trailing foot, yet there is a non-zero post impact velocity c . tr + ≠ 0 . In the second case, impulses are applied on each foot.

For these two cases, numerical values of the complete impulse vector Λ and the post-impact foot velocities c . + are shown as a function of φ (which ranges from 0 to 1 2 π ) in Figure 3. For the numerical evaluation, we used the minimalistic model (Figure 1(a)) and set φ . le − = φ . tr − . That is, we assume that the angle between the two legs is constant prior to touchdown. The results are normalized with respect to leg mass m, leg length l, and the pre-impact angular velocity of the trailing leg φ . tr − . Consequently, linear velocities are expressed in units of l φ . tr − and impulses in units of ml φ . tr − . To yield a positive value for φ . tr − (and thus positive values for the units of velocities l φ . tr − and impulses ml φ . tr − ), the model is walking towards the negative x -direction.

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Fig. 3.

Outcome of the collision computation as a function of the impact angle φ under the assumption of no sliding. Shown are post impact contact point velocities ( c . tr + , c . le + ) and impulses Λ for two scenarios: collision at the front foot while the back foot leaves the ground (case 1, solid lines) and collision of both feet (case 2, dashed lines). The results are normalized with respect to the leg mass m, the leg length l, and the pre-impact angular velocity φ . tr − . Case 1 can only explain the collision in the regions marked A and E. In between, the post-impact normal velocity of the trailing foot ( ẏ tr + ) becomes negative. Case 2 can only explain the collision in regions C, D, and E. For smaller angles, the normal impulse Λ tr , y becomes negative. In region B, neither case yields a consistent solution. In region E, the outcome is ambiguous, as both cases produce valid solutions.

For case 1 (solid lines), the results provide a consistent solution for pre-impact angles close to 0 and close to 1 2 π (regions marked as A and E). In these regions, the vertical post impact velocity of the trailing foot ẏ tr + is positive and the trailing foot will indeed leave the ground. In between, the vertical post impact velocity of the trailing foot ẏ tr + is negative. This contradicts the assumption that the foot leaves the ground after the collision. Case 1, lift-off at the trailing foot, is thus not a valid solution in regions B, C, and D.

Case 2 (dashed lines), on the other hand, provides consistent solutions with positive normal impulses for impact angles φ > 1 6 π (regions C, D, and E). For smaller angles (regions A and B), however, the normal impulse at the trailing foot Λ tr , y is negative. This contradicts the assumption that the trailing foot will remain in contact, since the unilateral ground contact can only produce positive normal forces.

Since case 1 only explains regions A and E, and case 2 only regions C, D, and E, the region marked B is left unexplained by either case. If we assume that the trailing foot leaves the ground, we obtain a normal velocity that would move the foot into the ground. If we assume that the trailing foot stays on the ground, we obtain a negative normal impulse that cannot be produced by a unilateral contact. Under the assumption of no slip, neither coming to a rest nor leaving the ground yields a valid solution.

3.2. Sliding with finite friction

A way to understand the apparent paradox of the previous section, is to realize that even with positive normal ground reaction forces Λ tr , y , the trailing foot would start to slide if the magnitude of the tangential component Λ tr , x were larger than μ Λ tr , y (where μ is the coefficient of friction). With μ = 1 , for example, sliding would occur as soon as the absolute value of the tangential force is larger than the normal force. For the chosen parameter values, this will happen in region C. In other words, under the assumption of finite friction with μ = 1 , case 2 will cease to be a valid solution in this range.

We can consider sliding as a third case of the collision process, for example by assuming a Coulomb impact at the trailing foot with a coefficient of friction μ (Figure 4):

Case 3. The trailing foot slides after the collision. Only the normal component of the trailing foot velocity will remain zero: ẏ tr + = 0 . The horizontal trailing foot velocity ẋ tr is unconstrained. For a Coulomb impact in which the trailing foot slides in the negative − x direction (i.e. in the direction of travel), the tangential trailing foot impulse Λ tr , x is given as Λ tr , x = μ Λ tr , y . For sliding in the positive x -direction, it is Λ tr , x = − μ Λ tr , y .

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Fig. 4.

When the impulse at the trailing foot moves to the edge of the friction cone, its tangential component Λ tr , x is bounded to be Λ tr , x = μ Λ tr , y . The foot is brought to a rest in the vertical direction, but sliding occurs in the horizontal direction.

Note that other formulations exist for the consideration of friction. Routh’s method, for example, applies the friction limit on a differential impulse ( d Λ x < μ d Λ y ) that is accumulated over the collision process (Wang and Mason, 1992). This method is applied, for example, in the work by Posa et al. (2016). For case 3, the resulting impulses and post impact velocities are shown in Figure 5. For μ = + 1 (dotted lines), the Coulomb impulse accurately describes the collision outcome in regions B and C. In this range, the normal impulse Λ tr , y is positive, and the post-impact velocity of the trailing foot ẋ tr + and the tangential impulse Λ tr , x have different signs. In regions A and E, the normal impulse at the trailing foot Λ tr , y is negative which violates the assumption that the foot undergoes a collision. In region D, the tangential component of the post-impact velocity ẋ tr + and the tangential impulse Λ tr , x have the same sign (both are positive), which violates the assumption of a Coulomb friction process.

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Fig. 5.

Outcome of the collision computation as a function of the impact angle φ , if the trailing foot slides after the collision (case 3). Shown are post impact foot velocities ( c . tr + , c . le + ) and impulses Λ for μ = + 1 (dotted lines) and μ = ∞ (long dashes). The results are normalized with respect to the leg mass m, the leg length l, and the pre-impact angular velocity φ . tr − . The bounded tangential impact ( Λ tr , x = μ Λ tr , y ) can correctly explain the collision outcome for leg angles in the region marked B for μ = ∞ , and in the regions marked B and C for μ = 1 .

Similarly, backwards sliding (with μ = − 1 ) provides a consistent solution for very large leg angles (not shown). In other words, as soon as sliding is allowed as a potential outcome, we can find valid solutions for all values of φ . In particular, sliding can explain the collision outcome for intermediate leg angles (region B) which had no solution under the simple assumption of no sliding.

3.3. Sliding with infinite friction

Even in the limit of μ → ∞ , the tangential impulse remains bounded and sliding will occur (see Appendix B.1 equations (18) to (23) for an analytical derivation). This limit of case 3 is shown in Figure 5 (long dashes). It produces a consistent solution for region B, which we were not able to explain by the simple assumption of no slip. In regions A, C, and D the tangential impulse Λ tr , x and the tangential component of the post-impact velocity ẋ tr have the same sign, which violates the assumption of a Coulomb friction impact. In region E, the assumption of sliding with infinite friction again yields consistent results and the trailing foot will slide in the positive x -direction.

Together, cases 1 to 3 can describe the complete touch-down behavior of the minimalistic model. As the impact leg angle φ increases, the system undergoes three structurally different outcomes: lift-off of the back foot, sliding of the back foot (even if we assume infinite friction), and coming to a complete stop. Sliding will always happen in region B, and depending on the magnitude of μ , this region will expand towards larger leg angles φ .

In addition, we saw that for large impact leg angles φ (region E), three outcomes are equally valid solutions. Coming to a stop, lifting off, and sliding all yield valid solutions if we assume that the coefficient of friction μ is infinitely large. This seems paradoxical at first, but must be understood as a limiting case of an extreme sensitivity that the dynamics exhibit with respect to initial conditions and parameters and that result from the approximation of the bodies being rigid. This is a common phenomenon in systems with unilateral contact. According to Stewart (2000), predicting for any system ‘… which of the possible solutions actually occurs requires knowledge of the microscopic details of the contact, which is not available at this level of modeling’. To summarize the collision outcomes, Figure 6 combines all three cases, highlighting a continuity of impulse values and post-impact velocities as the impact angle φ varies from 0 to 1 2 π . It also shows the three possible solutions in regions E.

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Fig. 6.

Combination of the results presented in Figures 3 and 5 for the case of μ = ∞ . Post impact foot velocities ( c . tr + , c . le + ) and impulses Λ are continuous as the impact angle φ varies. The system undergoes a lift-off of the trailing foot (in region A), sliding with infinite friction (region B), and a complete stop (regions C and D). For impact angles φ > 5 12 π , all three of these outcomes are valid solutions (region E).

3.4. Complete linear complementary problem formulation

In order to simplify the analysis to a two-dimensional contact space, we have so far assumed that the leading foot always comes to a complete stop. That is, c . le + = 0 . Throughout the analysis, we have not seen any indication that would contradict this assumption. For all values of φ , the normal impulse Λ le , y at the leading foot was positive. If the coefficient of friction is sufficiently large, the leading foot can be brought to a stop.

For smaller coefficients of friction, however, we must account for the possibility of sliding of the leading foot. Including this possibility leads to a total of 16 different outcomes. Linear complementary problem (LCP) formulations, as presented for example by Pfeiffer and Glocker (2000) or Stewart (2000), provide a means of handling such a large number of cases in a systematic framework. Adapting the approach by Pfeiffer and Glocker (2000, Chapter 8.8), for a coefficient of restitution ϵ = 0 (that is, only for the compression phase), we implemented an LCP formulation of the minimalistic model. The details of this implementation are presented in Appendix B.2.

In order to obtain all possible solutions, we solved the LCP by permutating through all possible cases (Murty and Yu, 1988), rather than relying on a more efficient solver that terminates after a solution has been found (such as Lemke’s algorithm). The solutions for μ = 1 are shown in Figure 7. As the pre-impact angle φ varies from 0 to 1 2 π , six structurally different outcomes arise: the trailing foot lifts off as the leading foot comes to a rest (region I); the trailing foot slides forward as the leading foot comes to a rest (region II); both feet remain at rest after the collision (region III); the trailing foot slides backwards as the leading foot comes to a rest (region IV); the trailing foot slides backward and the leading foot forward (region V); the trailing foot lifts off, and the leading foot slides forward (region VI). Multiple solutions do not arise at any of these configurations. This changes for larger coefficients of friction. For μ = 5 (Figure 8), there exists a region (labeled IV), in which sliding, lift-off, and coming to a rest are equally possible solutions for the behavior of the trailing foot. For very large μ , the numerical results converge towards the combined results for μ = ∞ shown in Figure 6.

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Fig. 7.

The result of the LCP formulation for μ = 1 shows how the minimalistic walker undergoes a large number of different contact scenarios when sliding is explicitly enabled. As the pre-impact angle φ varies, the following structurally different outcomes are possible (numerals refer to the shaded areas in the figure). I: the trailing foot lifts off as the leading foot comes to a rest. II: the trailing foot slides forward as the leading foot comes to a rest. III: both feet remain at rest after the collision. IV: the trailing foot slides backwards as the leading foot comes to a rest. V: the trailing foot slides backward and the leading foot forward. VI: the trailing foot lifts off, and the leading foot slides forward.

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Fig. 8.

Post impact foot velocities and impulses for μ = 5 obtained via a LCP formulation. The areas marked I–III and VI correspond to identical contact scenarios as those in Figure 7. Due to the larger coefficient of friction, however, multiple solutions are possible for pre-impact angles in the region marked as IV. These solutions are highlighted with different line styles. In region V, the trailing foot lifts off as the leading foot comes to a rest.

4. The reason for these behaviors

In order to facilitate a parametric analysis of the PDW model, we derived an explanation of these phenomena based on the relative angle of impulse vectors. It adds to the number of explanations that deal with related phenomena and that are primarily discussed in the context of Painlevé’s paradox (e.g. Brogliato, 1999, chapter 5.6) (Génot and Brogliato, 1999). In particular, we consider a simplified collision example in which a single body at rest is subject to an external impulse I (Figure 9). The body sits on a surface that only allows motion in the positive y -direction and the impulse is applied right at the point where the body makes contact with the surface. c . + is the velocity of the contact point after the impulse has been applied and Λ is the integral of the contact forces that arises from the unilateral constraint. The example in the figure shows a body with mass m=1 and inertia j = 0 . 5 and a COG that is located at (-1, 1) relative to the contact point. For this example, the contact space mass matrix M λ is

The integrated EOMs are:

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Fig. 9.

Example of a single rigid body in contact. The body is at rest when an external impulse I is applied at the contact point. Due to the off-diagonal terms in the contact space mass matrix M λ , the total impulse and the direction of c . + are not collinear. As a result, the collision outcomes can be ambiguous and sliding with infinite friction is possible. In the figure, α denotes the angle of inertial impulses as a reaction to pure sliding ( M λ ċ x + 0 ) and β the angle of the external impulse I . For α < β < 0 , the contact point can either lift-off (shown in (a)), stay in contact (shown in (b)), or slide (shown in (c)). For α + π < β < π , sliding can occur with infinite friction (shown in (d)).

In order for the contact point to slide along the contact surface after the impulse was applied (i.e. ċ y + = 0 ), the sum of the impulses Λ + I must be of the form

In other words, they must be directed along a line oriented at an angle of α = ∡ m x m d . ¹ For the contact point to slide in the negative x -direction, the sum of Λ and I must thus not only be directed towards the left, but also away from the contact surface. Similarly, to make the contact point slide in the positive x -direction, Λ + I must be directed towards the right and into the contact surface. If the angle β = ∡ I of the external impulse lies between the surface and the line with slope α , the following effects arise.

For the remaining ranges of the angle β , more intuitive outcomes happen. For 0 < β < α + π the contact point will always lift off, and for − π < β < α , the contact point will always stay in contact. The latter case can lead to either sliding or remaining at rest, depending on the coefficient of friction μ .

For a bipedal system (and under the assumption that the leading foot comes to a complete stop), equation (6) can be interpreted as being analogous to equation (7). The external impulse I is given by I = M cp c . le − and can be interpreted as a manifestation of the leading foot collision at the trailing foot. With regard to the remaining terms, the contact space mass matrix M λ is given by M tr , and the post-impact velocity c . + by the trailing foot velocity c . tr + .

5. Model of a passive dynamic walker

Based on these insights, we extend our investigation to the parametric analysis of the model of a PDW. PDWs are simple mechanical mechanisms that can walk down a shallow ramp, relying solely on gravity and their mechanical dynamics. Inspired by simple mechanical toys (Wilson, 1938), McGeer (1990a) was the first to build a functional prototype and to describe the concept mathematically. More complex versions have knees (McGeer, 1990b) or can move stably in three dimensions (Collins et al., 2001). PDWs are one of the most widely used models of legged locomotion since passive mechanical dynamics play an important role in human locomotion (Kuo, 2007; Mochon and McMahon, 1980). They have been used to explain, for example, the relationship of speed and stride frequency (Kuo, 2001), the investigation of stability and energetics for lateral motion (Bauby and Kuo, 2000; Donelan et al., 2001), and the effectiveness of active push-off to reduce collision costs (Kuo et al., 2005). In physical implementations, bipedal robots that are based on PDWs have shown unsurpassed levels of energetic economy (Bhounsule et al., 2014; Collins et al., 2005; Dertien, 2006; Wisse et al., 2007).

The model of the PDW used in this study (Figure 1(b)), differs from the conceptual model discussed in Section 3 in that the legs have rotational inertia j and that the COG can be placed arbitrarily. ² The COG position is given by the distance s ∥ along the line connecting the hip to the contact point and the distance s ⊥ perpendicular to this line. In this section, we investigate how these parameters influence the behavior of the system during collisions. As shown in Section 4, the type of the collision outcome depends only on the relative values of the angles α and β . Analytical expressions for these angles are derived in Appendix B.3. For any PDW, α will always be negative and will gradually vary from − π 2 to 0 as φ ranges from 0 to 1 2 π . Because α is negative, multiple solutions will occur for α < β < 0 and sliding with infinite friction will occur for α + π < β < π .

When we simplify the collision process to cases in which φ . le − = φ . tr − (which is not generally true, but often a close approximation in actual PDWs), the vector of the impulse I = M cp c . le − rotates from pointing vertically upwards towards pointing horizontally towards the right as φ ranges from 0 to 1 2 π . Depending on the system parameters, this rotation happens in a clockwise direction (for r g > s ∥ l ) or in a counter-clockwise direction (for r g < s ∥ l ). If the impulse vector rotates in a counter-clockwise rotation, there has to be a pre-impact angle range in which α + π < β < π . Sliding with infinite friction will occur in this range. In a clockwise rotation, β < α + π for all φ , and the conditions for sliding with infinite friction ( α + π < β < π ) are never fulfilled. With regards to multiple solutions, these can happen any time α < β < 0 . Such a region exists in the neighborhood of φ = 1 2 π for nearly all parameter choices. It vanishes only in the case of r g = s ∥ l . For r g > s ∥ l , an additional second region of multiple solutions exists at smaller pre-impact angles φ . The relative size of r g and s ∥ thus determines the occurrence of sliding with infinite friction and the occurrence of multiple solutions in models of PDWs.

Figure 10 shows the angles α and β as a function of φ for s ∥ = 1 3 l, s ⊥ = 0 l, φ . le − = φ . tr − , and for three different values of the leg inertia j. These parameters reflect the COG position that is depicted in the PDW in Figure 1(b). The first inertia value (shown in Figure 10(a)) is j = 5 36 ml² ( r g = 0.5 l). It approximates the inertia of the shown walker if its legs were made of homogeneous material of constant thickness. To visualize the effects discussed in this paper, the walker in Figure 1(b) is shown with an impact angle of φ = 1 10 π . For this impact angle, one can see in Figure 10(a) that α + π < β < π . That is, in the situation depicted in Figure 1(b), the trailing foot will slide, even for μ = ∞ . The second inertia value j = 2 9 ml² ( r g = 1 3 l) reflects the limiting case r g = s ∥ l (Figure 10(b)) and the third value j = 1 3 ml² ( r g = 2 3 l) shows how the structure of the solution changes if the leg inertia is increased beyond this critical value (Figure 10(c)).

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Fig. 10.

Shown are the angles α = ∡ M tr : , 1 and β = ∡ I (with I = M cp c . le − ) as a function of φ for the model of a PDW. The relative values of α and β determine the outcome of the collision process. For − π < β < α , the trailing foot will remain in contact (region marked as C and D). For α < β < 0 , multiple solutions are equally valid (region E). For 0 < β , α + π , the trailing foot will leave the ground (region A). For α + π < β < π , the trailing foot will slide, even for μ = ∞ (region B). Depending on the system parameters, not all of these outcomes happen. The three plots show results for different values of the leg inertia j: j = 5 36 ml² (shown in (a)) approximates the inertia of the walker shown in Figure 1(b) if its legs were made of homogeneous material of constant thickness. j = 2 9 ml² (shown in (b)) reflects the case r g = s ∥ l in which the regions of sliding with infinite friction and multiple solutions vanish. j = 1 3 ml² (shown in (c)) shows how the structure of the solution changes if the leg inertia is increased beyond this critical value. One can clearly see the change in the direction of rotation of the β angle. The remaining parameters are: s ∥ = 1 3 l, s ⊥ = 0 l, and φ . le − = φ . tr − .

6. Model of a bipedal robot

The same effects occur in more complex models of legged locomotion. For example, we applied the presented approach to a model of the bipedal robot RAMone (Figure 1(c)). RAMone is a typical planar five-link robot with non-articulated circular feet. Similar legged robots are described, for example, in the work by Chevallereau et al. (2005); Saglam and Byl (2013) and Westervelt et al. (2003). All states, dimensions, and mass properties of this robot are given in Table 1 and are illustrated in Figure 11.

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Table 1.

Dimensions and mass properties of the model of RAMone.

Param.	Value	Unit	Description
m 1	7.903	kg	Mass of main body
m 2	0.789	kg	Mass of upper leg segment
m 3	0.510	kg	Mass of lower leg segment
j 1	0.0800	kg m2	Inertia of main body
j 2	0.00221	kg m2	Inertia of upper leg segment
j 3	0.00652	kg m2	Inertia of lower leg segment
s 1	0.138	m	Distance of hip to main body COG
s 2	0.019	m	Distance of hip to upper leg COG
s 3	0.165	m	Distance of knee to lower leg COG
l 2	0.200	m	Length of upper leg segment
l 3	0.239	m	Length of lower leg segment
r foot	0.028	m	Foot radius

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Fig. 11.

Shown are the states (red) and model parameters (blue) of a model of the robot RAMone. The walker is pivoting about the trailing foot with all joint velocities being zero ( α . tr , le = β . tr , le = 0 ). In the shown configuration, the back leg will slide even in the presence of infinite friction.

Similar to the previous models, we considered a scenario in which the robot is pivoting about the trailing foot (which is at rest and located at x tr = y tr = 0 ) with an angular velocity of φ . tr − = 1 , while the hip and knee joints remain motionless ( α . tr , le = β . tr , le = 0 ). At the instance of the collision, both knees have an angle of β tr = β le = − 1 2 π and the main body is vertical ( ϕ = 0 ). This large knee bend is typical for a running gait of RAMone and having the knees point against the direction of motion increases the efficiency of locomotion (Smit-Anseeuw et al., 2017). Analogous to our prior analysis, we investigated the collision behavior of this system as a function of the inner leg angle φ . In this case, φ tr = − φ le = φ is defined with respect to a line connecting the hip joint to the foot (Figure 11). From these values, the hip angles α , the position of the main body (x, y), and the velocities of the main body ẋ , ẏ , ϕ . can all be computed given a particular choice of φ . We numerically computed the mass matrix M , the contact Jacobian J , and the contact space mass matrix M λ as a function of φ . ³ We then solved for the post impact velocities and the contact impulses for the three cases that were discussed in this paper, assuming an infinite coefficient of friction μ = ∞ .

As a consequence of the additional joints the impulse transfer from leading to trailing contact point is highly attenuated. Only very small ( < 1 0 − 4 Ns) impulses arise at the trailing contact point. Nonetheless, sliding with infinite friction and multiple valid solutions appear. In particular, there exists two bands of φ angles (0.47 rad < φ < 0.54 rad and 1.02 rad < φ < 1.06 rad) in which the trailing foot will slide despite infinite friction, and one band (1.34 rad < φ < 1.40 rad) in which sliding, coming to a stop, and lift-off are equally valid solutions. Table 2 gives the three collision outcomes for the example of an inner leg angle of φ = 0.5 rad, for which sliding with infinite friction occurs.. This is the configuration depicted in Figure 11. The table highlights the negative vertical post-impact velocity ẏ tr + if we assume that the trailing leg lifts off and the negative normal contact impulse Λ tr , y if we assume that the trailing leg comes to a complete stop. Sliding with infinite friction, however, is a valid solution as the post impact horizontal velocity ẋ tr + and the horizontal contact impulse Λ tr , x have opposite signs.

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Table 2.

Post impact velocities (in units of ms − 1 ) and contact impulses (in units of Ns) of the trailing leg of a model of RAMone are shown for the three collision cases with an inner leg angle φ = 0.5 rad. In this configuration q = x , y , ϕ , α tr , β tr , α le , β le T = − 0 . 149 , 0 . 439 , 0 , 1 . 373 , − 1 . 571 , 0 . 373 , − 1 . 571 T and q . − = − 0 . 439 , − 0 . 149 , 1 , 0 , 0 , 0 , 0 T . Sliding with infinite friction is the only viable outcome. This configuration is depicted in Figure 11.

	ẋ tr +	ẏ tr +	Λ tr , x	Λ tr , y
case 1	0.826 × 1 0 − 4	–0.032 × 1 0 − 4	0	0
case 2	0	0	–0.116 × 1 0 − 4	–0.043 × 1 0 − 4
case 3 ( μ = ∞ )	0.582 × 1 0 − 4	0	–0.033 × 1 0 − 4	0

7. Discussion and conclusion

In this paper, we investigated the details of the collision process of models of bipedal systems. In particular, we show that the well-established and widely used assumption of no sliding of the contact points can yield inconsistent results. For some contact configurations and system parameters, neither the assumption of lift-off of the trailing foot nor the assumption that it stays fixed on the ground yield valid solutions. If we assume that the trailing foot lifts off, we obtain post-impact velocities that would move the foot into the ground. If we assume that the trailing foot remains fixed, we obtain negative normal contact forces. To overcome these inconsistencies, sliding of the contact point is introduced as a third collision outcome. We showed that such sliding can occur even under the assumption of an infinite coefficient of friction. This is due to the fact that in the limit of μ → ∞ , the normal contact impulse goes to zero while the Coulomb friction process yields a finite tangential impulse. This impulse is not necessarily large enough to bring the tangential velocity to a complete halt.

In a related effect, we discovered situations in which the solution of the collision process is ambiguous. If the coefficient of friction is large enough, all three cases (i.e. the trailing foot remaining at rest, lifting off, and sliding) result in mathematically valid solutions. For a rigid-body-model of a biped that undergoes an impulsive collision process, the state after the collision cannot always be determined uniquely.

These non-intuitive behaviors in models of bipedal systems were presented for a minimalistic biped, a passive dynamic walker, and a model of the five-link biped RAMone. Via the simplified scenario of a single body in unilateral contact that is subject to an external impulse, the effect was further examined. We showed that the occurrence of the above-mentioned behaviors depends on the relative direction of the external impulse and the reaction force to a purely tangential acceleration. Based on these insights, and under the assumption that the trailing and leading leg rotate with the same angular velocity prior to impact, we derived ranges of system parameters in which sliding with infinite friction and multiple valid solutions are possible for the passive dynamic walker.

The ambiguous outcomes with more than one valid solution were only observed in contact configurations with very large leg angles φ (when the robot is close to doing the splits) or for very large leg inertia j (in which the mass is concentrated at or below the foot). From a practical perspective, these results play probably only a minor role in the study of legged locomotion. The occurrence of sliding, even when infinite friction is assumed, has more direct implications. For boththe RAMone model and the PDW, sliding of the trailing leg was shown to happen in a range of leg angles φ and inertia values j that are very reasonable for walking systems. In fact, the PDW and RAMone configurations shown in Figures 1(b) and 11, depict situations in which sliding with infinite friction would occur. Similarly, the minimalistic model entered a sliding state (with infinite friction) for 1 12 π ≤ φ ≤ 2 12 π . For comparison, the leg angle φ during human walking is about 0 . 38 rad ≈ 1 . 5 12 π (Grieve, 1968); that is, the presented effect happens for leg configurations that are well in the range of regular walking.

Due to the large number of parameters and states, we refrained from conducting a similarly detailed parametric and state dependent analysis for the model of RAMone as we have done it for the PDW. Instead, we limited our analysis to merely showing that the effect does arise in a very basic configuration of this realistic model. When playing with the configurations of the robot model, it did however become apparent that the outcomes were quite sensitive to changes in states and velocities. For example, pitching the main body forward and backward by ± 0 . 25 rad shifted the band of angles φ in which sliding with infinite friction happened by about ± 0 . 1 rad. This variability highlights that the observed effects are not necessarily limited to a set of particular pre-impact configurations (such as, for example, configurations close to kinematic singularities). Instead, they can arise over a range of different configurations, creating a certain unpredictability in more complex models. So while the effects were limited to a very small range of pre-impact configurations in the model of RAMone, it might be difficult to exclude and ignore them completely. In particular, it seems that one cannot simply evade them by simply staying away from particular regions of the configuration space.

In the passive models, sliding and coming to a complete stop both represent situations that interrupt locomotion. PDWs can, by definition, only walk when the trailing foot leaves the ground passively after the collision. This requirement limits their operation to leg angles φ in the region marked A in Figures 3, 5, and 6 and as region I in Figures 7 and 8. For passive systems, it is thus sufficient to check the vertical post-impact velocity of the trailing foot to detect a gait failure. In contrast, actuated walkers can rely on an active push-off and thus might deliberately enter a phase of double-support after touch-down of the leading foot. For active systems, the different collision outcomes must not only be detected but also properly processed. In this context, one needs to be aware that sliding can never be fully excluded as an outcome, even in idealized models with infinite stiction. In forward dynamic simulations, the most concise way of handling the possible outcomes is through an LCP formulation, as it is commonly used in time-stepping algorithms (Anitescu and Potra, 1997; Glocker, 1995; Stewart, 2000; Stewart and Trinkle, 1996). When solving these LCPs with efficient algorithms, care should be taken to ensure that all solutions of the LCP can be found and that the occurrence of ambiguous solutions can be detected.

For more complex articulated models, the inconsistencies can be fairly small. For the model of RAMone, for example, the post-impact velocity or contact impulse were only in the order of 1 0 − 6 for the range of φ in which sliding with infinite friction happened (Table 2). This is because the contact impulses get attenuated in the articulation of the leg. If the effect is limited to such small values, one could also chose to deliberately ignore the effect and, for example, circumvent the case of sliding with infinite friction by allowing a negative impulse whenever a negative ẏ tr + was found.

By studying bipedal models that range from minimalistic to physically accurate, this work establishes a bridge between the theoretical study of unilateral contact in multibody dynamic simulations and the application of these simulation methods in the modeling of ground contact in actual legged systems. We highlight, examine, and explain the details of the collision process and investigate in particular the non-intuitive behaviors of sliding with infinite friction and the simultaneous occurrence of multiple valid solutions. By showing these effects on the multi-body model of an actual five-link walker, we emphasize their relevance to the field of legged robotics.