Passivity-Based Control of Bipedal Locomotion
IEEE robotics & automation magazine
T his article is an overview of our recent results in passivity-based control in biped locomotion. The article represents a synthesis of ideas from  , - into a set of cohesive results that exploits the notion of passive walking within the context of hybrid passivity-based control to achieve regulated walking on varying slopes, robustness to uncertainties and disturbances, as well as to regulate walking speed and gait transitions. The idea of passive dynamic walking, pioneered by
... er more than a decade ago  , has been well studied by several researchers -,  and will not be discussed here. We are interested primarily in active control methods that exploit the existence of passive gaits in two-dimensional (2-D) and three-dimensional (3-D) bipeds. While passive dynamic walking is appealing for its elegance and simplicity, active feedback control is necessary to achieve walking on level ground and varying slopes, robustness to uncertainties and disturbances, and to regulate walking speed. In this article, we show how to achieve these properties within the context of passivity-based control. The first results in active feedback control that exploit passive walking appeared in  , , ,  for planar bipeds. Passive walking in three dimensions was studied in  and . Later, the results in  were extended to the general case of 3-D walking in . An interesting and elegant extension of these ideas appears in  where geometric reduction methods are used to generate stable 3-D walking from 2-D gaits. Robustness issues were addressed in  using total energy as a storage function in the hybrid passivity framework. In  it was shown how speed regulation, In this section, we discuss the dynamics of bipedal locomotion. Our treatment will be necessarily brief as the hybrid Lagrangian model treated here has been developed and used extensively by numerous other researchers , , , . See, for example,  for a detailed derivation of a general model in the planar case. The act of walking involves both a swing phase and a stance phase for each leg as well as impacts between the swing leg and ground, and possibly internal impacts, such as knee strikes due to mechanical constraints on the joints. Consider an n degree-of-freedom (DoF) biped during the single-support phase as shown in Figure 1 . Each joint of the robot is assumed to be revolute and to allow a single DoF rotation. The stance leg, which is in contact with the ground, has three DoF relative to an inertial frame (assuming no slipping). We can therefore use Q = SO(3) × T n−3 to represent the configuration space of the biped, where SO(3) is the rotation group in R 3 and T n−3 is the (n − 3)-torus. A configuration is then characterized by an ordered pair q = (R, r), where R ∈ SO(3) is the orientation of the first link and r ∈ T n−3 is the shape of the multibody chain, for example, the angle of each joint referenced to the previous joint. Given a configuration, q = (R, r) ∈ SO(3) × T n−3 , we represent a velocity vector in T q Q via the pair (R −1Ṙ ,ṙ) ∈ so(3) × R 3 , where so (3) is the Lie Algebra of 3 × 3 skew-symmetric matrices. The advantage of this formalism is that only the orientation of the first link (actuated by the stance ankle) is referenced to an absolute or world frame. The remaining n − 3 joint variables, called the shape variables, are then invariant under a change of basis of the world frame. Configuration spaces that can be written as the Cartesian product of a Lie group and a shape space are referred to as principal bundles; see  .