During locomotion, when the swing limb (i.e. the limb that is not on the ground) contacts the ground surface (heel strike), the generalized velocities will be subject to jump discontinuities resulting from the impact event. Also, the roles of the swing and the stance limbs will be exchanged, resulting in additional discontinuities in the generalized coordinates and velocities [15].
The individual joint rotations and velocities do not actually change as the result of switching. Yet, from biped's point of view, there is a sudden exchange in the role of the swing and stance side members. This leads to a discontinuity in the mathematical model. The overall effect of the switching can be written as the follows:
x* = Sw x (2.3)
where the superscripts x* is the state immediately after switching and the matrix Sw is the switch matrix with entries equal to 0 or 1.
Using the principles of linear and angular impulse and momentum, we derive the impact equations containing the impulsive forces experienced by the system. However, applying these principles require some prior assump- tions about the impulsive forces acting on the system during the instant of impact. Contact of the tip of the swing limb with the ground surface initiates the impact event. Therefore, the impulse in the y direction at the point of contact should be directed upward. Our solution is subject to the condition that the impact at the contact point is perfectly plastic (i.e. the tip of the swing limb does not leave the ground surface after impact). A second under- lying assumption is that the impulsive moments at the joints are negligible.
When contact takes place during the walking mode, the tip of the trailing limb is contacting the ground and has no initial velocity. This is always true when the motion is no-slip locomotion. The impact can lead to two possible
110 Y. Hurmuzlu
outcomes in terms of the velocity of the tip of the trailing limb immediately after contact. If the subsequent velocity of the tip in the y direction is pos- itive (zero), the tip will (will not) detach from the ground, and the case is called "single impact" ("double impact"). We identify the proper solution by checking a set of conditions that must be satisfied by the outcome of each case (see Fig. 2.1).
L/ ~ O o ~ D l ~ ,
/ //
Contact
Before I m p a c t
L
After Impact
Fig. 2.1. Outcomes of the impact event Solution of the impact equations (see [20] for details) yields:
x + = I r a ( x - ) (2.4)
where x - and x + are the state vector before and after impact respectively, and the matrix Im(X--) is the impact map.
2.3 S t a b i l i t y o f t h e L o c o m o t i o n
In this chapter, the approach to the stability analysis takes into account two generally excepted facts about bipedal locomotion. T h e motion is discontin- uous because of the impact of the limbs with the walking surface [15, 18, 28].
T h e dynamics is highly nonlinear and linearization about vertical stance should be avoided [17, 27].
Given the two facts that have been cited above we propose to apply discrete m a p p i n g techniques to study the stability of bipedal locomotion. This approach has been applied previously to study of the dynamics of bouncing
Dynamics and Control of Bipedal Robots 111 ball [8], to the study of vibration dampers [24, 25], and to bipedal systems [16].
The approach eliminates the discontinuity problems, allows the application of the analytical tools developed to study nonlinear dynamical systems, and brings a formal definition to the stability of bipedal locomotion.
The m e t h o d is based on the construction of a first return map by con- sidering the intersection of periodic orbits with an k - 1 dimensional cross section in the k dimensional state space. There is one complication that will arise in the application of this method to bipedal locomotion. Namely, dif- ferent set of kinematic constraints govern the dynamics of various modes of motion. Removal and addition of constraints in locomotion systems has been studied before [11]. T h e y describe the problem as a two-point boundary value problem where such changes may lead to changes in the dimensions of the state space required to describe the dynamics. Due to the basic nature of discrete maps, the events that occur outside the cross section are ignored.
T h e situation can be resolved by taking two alternative actions. In the first case a mapping can be constructed in the highest dimensional state space that represents all possible motions of the biped. When the biped exhibits a mode of motion which occurs in a lower dimensional subspace, extra di- mensions will be automatically included in the invariant subspace. Yet, this approach will complicate the analysis and it may not be always possible to characterize the exact nature of the motion. An alternate approach will be to construct several maps that represent different types motion, and attach var- ious conditions that reflect the particular type of motion. We will adopt the second approach in this chapter. For example, for no slip walking, without the double support phase, a mapping P h i l , is obtained as a relation between the state x immediately after the contact event of a locomotion step and a similar state ensuing the next contact. This map describes the behavior of the intersections of the phase trajectories with a Poincar5 section ~n~t~ defined
a s
_-- _-- < #,
< u, Fry > 0},
(2.5)
where XT and YT are the x and y coordinates of the tip of the swing limb respectively, # is the coefficient of friction, and F and F are ground reaction force and impulse respectively. The first two conditions in Eq. (2.5) establish the Poincar~ section (the cross section is taken immediately after foot con- tact during forward walking), whereas the attached four conditions denote no double support phase, no slip impact, no slippage of pivot during the single support phase and no detachment of pivot during the single support phase respectively. For example, to construct a map representing no slip running,
112 Y, Hurmuzlu
the last condition will be removed to allow pivot detachments as they nor- mally occur during running. We will not elaborate on all possible maps that m a y exist for bipedal locomotion, but we note that the approach can address a variety of possible motions by construction of maps with the appropriate set of attached conditions.
T h e discrete map obtained by following the procedure described above can be written in the following general form
(~ : P ( ~ - I ) (2.6)
where ~ is the n-1 dimensional state vector, and the subscripts denote the ith and (i - 1)th return values respectively.
0.6
+
¢ 5 0.3
0.0 i . 0.9 ~ :",,
l ,.~t~ ~ h . }~,t:-'~ ~7.=,-=~,:~:_ _~,----" .~; ....
W~?-" ~ l ... . . . . - 0 , 3 I I I lli,,~ = ~ ' " ¢ I I i ~ ' : - - . - i = ~ "
-0.6
-2.0 -1.6 -1.2 -0.8 -0.4 0.0
Step Length P a r a m e t e r
Fig. 2.2. Bifurcations of the single impact map of a five-element bipedal model
Periodic m o t i o n s of the biped correspond to the fixed points of P where
~, _ - p k ( ~ , ) . (2.7)
where p k is the kth iterate. The stability of pk reflects the stability of the corresponding flow. The fixed point ~* is said to be stable when the eigen- values v{, of the linearized map,
6~{ = D P k ( ~ *) 6~{-1 (2.8)
Dynamics and Control of Bipedal Robots 113 have moduli less that one.
This method has several advantages. First, the stability of gait now con- forms with the formal stability definition accepted in nonlinear mechanics.
The eigenvalues of the linearized map (Floquet multipliers) provide quanti- tative measures of the stability of bipedal gait. Finally, to apply the analysis to locomotion one only requires the kinematic data that represent all the relevant degrees of freedom. No specific knowledge of the internal structure of the system is needed.
The exact form of P cannot be obtained in closed form except for very special cases. For example, if the system under investigation is a numerical model of a man made machine, the equations of motion will be solved nu- merically to compute the fixed points of the map from kinematic data. Then stability of each fixed point will be investigated by computing the Jacobian using numerical techniques. This procedure was followed in [4]. We also note that this mapping may exhibit a complex set of bifurcations that may lead to periodic gaits with arbitrarily large number of cycles. For example, the planar, five-element biped considered in [12] leads to the bifurcation diagram depicted in Fig. 2.2 when the desired step length parameter is changed.
3. C o n t r o l o f B i p e d a l R o b o t s 3.1 A c t i v e C o n t r o l
Several key issues related to the control of bipedal robots remains unresolved.
There is a rich body of work that addresses the control of bipedal locomotion systems. Furusho and Masubuchi [5] developed a reduced order model of a five element bipedal locomotion system. They linearized the equations of motion about vertical stance. Further reduction of the equations were performed by identifying the dominant poles of the linearized equations. A hierarchical con- trol scheme based on local feedback loops that regulate the individual joint motions was developed. An experimental prototype was built to verify the proposed methods. Hemami et al. [11, 10] authored several addressing control strategies that stabilize various bipedal models about the vertical equilibrium.
Lyapunov functions were used in the development of the control laws. The stability of the bipeds about operating points was guaranteed by constructing feedback strategies to regulate motions such as sway in the frontal plane. Lya- punov's method has been proved to be an effective tool in developing robust controllers to regulate such actions. Katoh and Mori [18] have considered a simplified five-element biped model. The model possesses three massive seg- ments representing the upper body and the thighs. The lower segments are taken as telescopic elements without masses. The equations of motion were linearized about vertical equilibrium. Nonlinear feedback was used to assure asymptotic convergence to the stable limit cycle solutions of coupled van der Pol's equations. Vukobratovic et al. [27] developed a mathematical model to
114 Y. Hurmuzlu
simulate bipedal locomotion. The model possesses massive lower limbs, foot structures, and upper-body segments such as head, hands etc.; the dynamics of the actuators were also included. A control scheme based on three stages of feedback is developed. The first stage of control guarantees the tracking in the absence of disturbances of a set of specified joint profiles, which are partially obtained from hmnan gait data. A decentralized control scheme is used in the second stage to incorporate disturbances without considering the coupling effects among various joints. Finally, additional feedback loops are constructed to address the nonlinear coupling terms that are neglected in stage two. The approach preserves the nonlinear effects and the controller is robust to disturbances. Hurmuzlu [13] used five constraint relations that cast the motion of a planar, five-link biped in terms of four parameters. He analyzed the nonlinear dynamics and bifurcation patterns of a planar five- element model controlled by a computed torque algorithm. He demonstrated that tracking errors during the continuous phase of the motion may lead to extremely complex gait patterns. Chang and Hurmuzlu [4] developed a robust continuously sliding control scheme to regulate the locomotion of a planar, five element biped. Numerical simulation was performed to verify the ability of the controller to achieve steady gait by applying the proposed con- trol scheme. Almost all the active control schemes often require very high torque actuation, severely limiting their practical utility in developing actual prototypes.
3.2 Passive C o n t r o l
M c G e e r [21] introduced the so called passive approach. H e demonstrated that simple, unactuated mechanisms can ambulate on d o w n w a r d l y inclined planes only with the action of gravity. His early results were used by re- cent investigators [6, 7] to analyze the nonlinear dynamics of simple models.
T h e y demonstrated that the very simple model can produce a rich set of gait patterns. These studies are particularly exciting, because they demonstrate that there is an inherent structural property in certain class of systems that naturally leads to locomotion. O n the other hand, these types of systems cannot be expected to lead to actual robots, because they can only perform w h e n the robot motion is assisted by gravitational action. These studies may, however, lead to the better design of active control schemes through effective coordination of the segments of the bipedal robots.
4. O p e n P r o b l e m s a n d C h a l l e n g e s i n t h e C o n t r o l o f B i p e d a l R o b o t s
O n e w a y of looking at the control of bipedal robots is through the limit cycles that are formed by parts of d y n a m i c trajectories and sudden phase
Dynamics and Control of BipedM Robots 115 action of gravity may lead to proper impacts and switches in order to produce steady locomotion. Active control schemes are generally based on trajectory tracking during the continuous phases of locomotion. For example, in [12, 4], the motion of biped during the continuous phase was specified in terms of five objective functions. These functions, however, were tailored only for the single support phase (i.e. only one limb contact with the ground). The con- trollers developed in these studies were guaranteed to track the prescribed trajectories during the continuous phases of motion. On the other hand, these controllers did not guarantee that the unilateral constraints that are valid for the single support phase would remain valid throughout the motion. If these constraint are violated, the control problem will be confounded by loss of controllability. While the biped is in the air, or it has two feet on the ground, the system is uncontrollable [2]. To overcome this difficulty, the investiga- tors conducted numerical simulations to identify the parameter ranges that lead to single support gait patterns only. Stability of the resulting gait pat- operation. Developing general feedback control laws and stability concept for hybrid mechanical systems, such as bipedal robots remains an open prob- lem [2, 3].
A second challenge in developing controllers for bipeds is minimizing the required control effort in regulating the motion. Studying the passive (un- actuated) systems is the first effort in this direction. This line of research is still in its infancy. There is still much room left for studies that will explore the development of active schemes that are based on lessons learned from the research of unactuated systems [6, 7]. lems with friction. Several definitions of the coefficient of restitution have been developed: kinematic [22], kinetic [23] and energetic [26]. In addition, algebraic [1] and differential [19] formulations are being used to obtain the
116 Y. Hurmuzlu
equations to solve the impact problem. Various approaches m a y lead to sig- nificantly different results [20]. T h e final chapter on the solution of the impact problems of kinematic chains is yet to be written. Thus, modeling and control of bipedal machines would greatly benefit from future results obtained by the investigators in the field of collision research.
Finally, the challenges t h a t face the researchers in the area of robotics are also present in the development of bipedal machines. Compact, high power a c t u a t o r s are essential in the development of bipedal machines. Electrical mo- tors usually lack the power requirements dictated by bipeds of practical util- ity. G e a r reduction solves this problem at an expense of loss of speed, agility, and the direct drive characteristic. Perhaps, pneumatic actuators should be tried as high power a c t u a t o r alternatives. T h e y m a y also provide the compli- ance t h a t can be quite useful in absorbing the shock effect t h a t are imposed on the system by repeated ground impacts. Yet, intelligent design schemes to power the p n e u m a t i c actuators in a mobile system seems to be quite a challenging task in itself. Future considerations should also include vision systems for terrain m a p p i n g and obstacle avoidance.
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