The Stability Module

In comparison to robots most insects have the ability to cling to the ground using various techniques (Bässler1983; Bullock et al. 2008). This, however, works only if the ground is sufficiently solid—on loose sand they cannot hold to the ground. But even if the insect looses contact, due to the small drop height the impact will not be impairing. Therefore, insects do not have to make sure at all times that static stability is maintained. The typical robot, however, must avoid crashes at all costs. Since only few walking robots are equipped with adhesive pads or claws (for counterexamples see Murphy et al. 2011; Parness et al. 2013), they must ensure dynamic or static stability irrespective of the substrate. As rather slow walker, HECTOR relies on maintenance of static stability. Therefore, a dedicated module is required to supervise the stability of the system. To quantify the stability, a stability measure is used that estimates the energy that would be required to let the robot tumble (see section 5.5.1). Based on this measure, during walking, the stability module interacts with the other modules of the walking controller, e.g., by preventing leg liftoffs that might result in instability (see section5.5.2).

5.5.1. Measure for the Evaluation of Static Stability

Different measures of stability (stability margins) have been proposed to quantify the tendency of a system to topple over. The first stability margin was proposed by McGhee and Frank (1968) for a machine with constant velocity walking on flat, even terrain. In this concept, the stability margin equals the shortest distance between the vertical projection (along the vector of gravity) of theCOMto any point of the support polygon. Therefore, a big distance between the projection of the COMand the support polygon implies a high stability. For the application on a robot that will change speed and direction during walking and furthermore is supposed to walk in rough terrain this stability measure is not adequate since it does not take the vertical position of theCOM into account. The higher the COMis located above the support polygon, the more it resembles an inverted pendulum. Therefore, even slight disturbances can be sufficient to destabilize the robot. This dependence of the stability on the relative location of the COM is considered in the ESM (Messuri 1985, see below), which represents the minimum energy that is required to let a robot tilt around one of the lines spanning the support polygon. Although this stability measure was developed for stiff robots, due to the lack of an adequate stability measure that considers compliance, it will be used for the control ofHECTOR. In the next paragraphs, first, the concepts of the ESM will be described, followed by a discussion of the limits of this stability margin for the application in a compliant system.

stiff robot (ESM) As mentioned, the ESM (Energy Stability Margin) represents the minimum energy that is needed to let a robot tilt around one of the support polygon lines. In fig. 5.24, an exemplary system is depicted that consists of the two points p1

p1 p2 pCOM pT e3 e1 e2 p‘COM g

Figure 5.24.: Depiction of the computation of the stability margin. The points p1 and p2 correspond to two foot points that span the support polygon. pCOM is the actual position of the COM, for which the stability margin should be computed. p′_COM represents the virtual position of the COM at the highest point during a rotation about the line between p1 and p2. The stability margin is proportional to the energy required to lift theCOMfrom pCOM to p′COM.

and p2 of the support polygon and the COM that is located at pCOM. The minimal energy that is required to turn the mass around the axis p1p2 equals the difference of the potential energy between the current state (mass at position pCOM) and the maximal potential energy the body could reach while the mass is rotated around the axis. The point at which the highest potential energy is reached, will be denoted p′

COM. To compute the position of p′

COM, the vector e3 (see fig. 5.24) must be determined that is part of an orthogonal system of unit vectors e1, e2, e3. e3 lies in the plane spanned by e1 = (p2− p1)/kp2− p1k and the vector of gravity eg = _kgkg . It can be constructed

by e3 = e1× e2 with (5.67) e2 = − eg× e1 |eg× e1| . (5.68)

The closest point on the axis p1p2 around which theCOM is revolved, is called pT. It can be computed by an orthogonal projection of pCOM onto p1p2:

pT =

(pCOM− p1) · (p2− p1) kp2− p1k

(p2− p1) + p1 (5.69)

The distance between pT and pCOM is d = kpCOM− pTk. Thus, the position of the point of highest potential energy is p′

COM= pT+ e3· d. The difference of the potential energy between the state in which the center of gravity is at pCOM and the state where it is at p′

COMis

∆E = −(p′

COM− pCOM) · g m . (5.70)

This is the minimal energy that is needed to let the robot topple over the connection p₁p₂. As mentioned, theESM is defined as the minimal energy that is needed to let the robot tumble about any of the support polygon lines:

SESM= min ({∆Ei}i=1...n) (5.71)

with n being the number of lines of the support polygon and Ei being the energy that

is needed to tilt the center of gravity around a particular line.

For static situations, when the robot is rigidly standing, SESM≥ 0 denotes a stable and

SESM < 0 an instable posture. Although in principle only stable and instable situations must be distinguished, the stability margin is usually interpreted as a continuous measure of stability. This is only sensible if additional, unforeseeable factors must be taken into account. Therefore, a high stability margin means that the robot is more likely to maintain stability without intervention after an impact (e.g., a researcher kicking the robot). In general, the energy Eimp that is introduced into the robotic system due to the impact must be smaller than SESM to guarantee stability. Besides external disturbances, self-induced accelerations of the robot must be considered. Therefore, if the robot needs to stop immediately, e.g., due to an obstacle, the inertia of the robot might be sufficient to overcome SESM, letting the robot topple over. Thus, if sudden stops must be expected, the kinetic energy of the robot Ekshould be considered as the lower boundary of stability (Ek< SESM).

compliant robot (not solved) The computation of the energies required to tilt the robot about the lines of the support polygon is based on the assumption of a rigid robot. HECTOR, however, is equipped with compliant joints that would result in lowering of the COMduring the tilting motion. This poses a problem that has not yet been solved. In fig.5.25, the subsiding is depicted for standing and tilted postures. The potential energy is reduced in both cases for the compliant robot since the COM is lowered compared to the stiff robot. As a consequence, the potential energies of the springs in the joints would need to be considered as well to quantify the change of potential energy due to the tilting:

∆Ecomp= −(p′COM− pCOM) · g m +

N X i=1 E_spr,i′ ₋ N X i=1 Espr,i ! (5.72)

Espr,i and Espr,i′ represent the potential energy of the i-th spring in the standing and the tilted postures, respectively.

(a) (b)

Figure 5.25.: Schematic comparison of stiff (gray) and compliant (black) robots for two different postures. (a) shows robots during normal standing. The central body of the compliant robot is lowered due to the weight of the central body which loads the springs. (b) shows the robots in a critical posture, in which theCOMis situated vertically above the foot point. Again, for the compliant robot the position of the COMis lowered as compared to the stiff robot and the springs are loaded.

For the compliant joints used in HECTOR, the potential energy due to a torsion θ of the elastomer coupling can be approximated based on eq. (2.1), which results in:

Espr(θ) =

Z θ

ω=0τ dω (5.73)

= 6162.59θ6_{+ 32.025θ}2 _(5.74)

With the sensor data of the joint drives, the total potential energy stored within the elastomer couplings can be estimated. In the tilted posture all but two legs can be assumed to be unloaded. Therefore, the energy previously stored in the couplings may have facilitated the tilting of the robot. The missing values to compute ∆Ecomp (according to eq. (5.72)) are the position of the COM in the tilted posture and the corresponding torsions of the elastomer couplings in the loaded legs. The estimation of these values, however, would require extensive simulations. Nagy (1992) suggested a stability measure that incorporates the compliance of the terrain. Since it assumes a single spring-like compliance between the leg tips and the ground it cannot adequately represent the distributed compliance in the legs of HECTOR. A stability margin to quantify the static stability of compliant robots is therefore lacking.

combination with the ESM for the control of HECTOR. The stability margin is com- puted under the assumption of stiff legs (see eq. (5.70)), however, based on the actual, subsided posture of the compliant robot. This measure is used for the computation of the stability unrestrictedness (see section5.2.1) as well as for the mechanisms that will be introduced in section 5.5.2. During the performed experiments the robot remained stable insofar that it did not topple. Still, due to the simplifications general stability cannot be guaranteed for the compliant system.

5.5.2. Interaction During Walking

In the original concept of WALKNET, the legs inevitably switched to swing phase if they

reached theirPEP. However, as shown in chapter4, this might compromise the stability of the robot. To facilitate stable walking the stability module interacts with the walking controller in two ways, the pre-liftoff stability check and the stability maintenance.

pre-liftoff stability check Before the transition of a leg from stance to swing phase is performed, the prospective posture is checked by the stability module. For two cases, this is depicted in fig. 5.26. Assuming the leg would be released from stance phase, the stability margin is computed under consideration of the remaining stancing legs. If this prospective stability margin lies below a given threshold the leg controller is forced to remain in stance mode. Thus, as long as the leg is essentially required to maintain stability, it is appointed to stance. This mechanism can be regarded as an additional, artificial coordination influence since it prolongs the stance phase of individual legs.

Without the intervention of the pre-liftoff stability check, the legs would change to swing phase at the latest when they reach the end of the stance unrestrictedness. However, since this mechanism keeps the legs in stance phase, they might leave their workspace. To circumvent this, the velocity of the robot must be reduced if a leg gets close to its workspace limit. This mechanism is explained in more detail in section5.6.

stability maintenance In addition to the pre-liftoff stability check, the stability mar- gin is continuously checked to ensure stability. If the COM moves towards one of the lines spanning the support polygon, the movement must be stopped before the stability threshold is underrun. As previously mentioned, a gradual deceleration is preferable. The mechanism that predicts potential problems in advance and then reduces the speed of the robot is described in section5.6.