Passive Dynamic Walkers

A passive dynamic walker (PDW) is an entirely mechanical device that is able to exhibit a steady and stable gait down an declined slope purely due to gravitational forces and no other energy input. The energy gained by its progression down a slope due to gravity is lost during inelastic collision events such as knee strike (knee lock) or the heel strike. Its total energy remains constant throughout its gait cycle, hence it is completely passive.

An attractive aspect of a PDW to the field of gait analysis is that it allows us to separate the purely mechanical attributes of walking from the cognitive controls of the human body. This characteristic is advantageous when it is desired only to study the physical parameters of human gait such as step length, swing time, gait asymmetries, etc.

It is quite different from humanoid robots in its dynamics. All humanoid robots either follow a quasi-static pattern and/or require controllers to model feedback laws. Neither approach is analogous to human gait since human walking is dynamically stable and robot controllers do not yet adequately model the human sensorimotor system. A dynamically stable passive model is more realistic of the natural human gait dynamics and can predict the motions from altered dynamics. Toyota’s ASIMO [176], Aldebaran Robotics’s NAO [5], and Albert-HUBO [146] robots are statically stable robots that are able to simulate a slow and careful walking pattern while always keeping their center of gravity above their support base. Humans can walk this way, but rarely do. While more elegant and proficient in its gait, Google’s/Boston Dynamics’s Atlas [155] is an anthropomorphically correct biped able to mimic gait very similar to humans. Atlas is able to skillfully navigate across obstacles such as stairs and withstand moderate perturbations during gait.

However, in contrast to Atlas, PDWs are much simpler, and versatile, while retaining human-like dynamics while Atlas follows highly controlled dynamic algorithms. The mentioned humanoid robots can be seen in Figure 2.9

The two-dimensional PDW concept was pioneered by analyzing a rimless wheel pro-gressing down a slope, then developed into a inverted double pendulum model (compass gait).

The compass gait PDW model is represented by two straight legs progressing down a decline.

Figure 2.9: Modern humanoid robots. (Left-Right) Toyota ASIMO, Aldebaran Robotics’s NAO, Albert-HUBO, and Google’s (Boston Dynamics) Atlas. (All images in public domain)

Subsequently, that model was advanced into a kneed walker modeled by two system phases during limb swing. By differentiating left and right legs and varying leg mass and mass distribution, Honeycutt et al. [87] enabled an asymmetric PDW while also adding additional masses to the PDW model. Two or more masses per link allows the first and second moment of inertia to be uncoupled, which yields more versatility in distributing the lumped masses along the link. These evolutionary steps in PDW development were modeled with one or no masses per link as seen in Figure 2.10)

As previously discussed, roll over shapes (ROS) have a great impact on gait dynamics, balance, and control. Constant radius ROS have also been included into PDW models, playing a crucial role in the design of passive dynamic walkers (PDW). Through design trials, McGeer indicates a most effective foot rocker radius to be 1/3 of total leg length [132], exactly matching

Figure 2.10: PDW evolution. (a) Rimless wheel (b) Compass Gait (c) Three-link (Kneed) walker

the most efficient human ROS radius [1]. Wu et al. [215] used a method of trial-and-error to design the curved feet of a PDW ROS, while emphasizing the importance of ROS in the design of PDWs, stating: "More information is needed about the effect of the foot roll-over shape on the allowable size of the disturbances". Although PDW ROS are a key component to the dynamics and stability of PDWs, currently I am not aware of any literature that specifically studies and specifies the size or shape of PDW ROS. Also, I was not able to find any research which studies the effect of asymmetric ROS feet or ROS with given continuous functions (step, trigonometric, etc.). I was only able to find one study which uses a compass gait PDW, modeling its ROS in a discrete and numerical mannor and deviding the ROS into two sections, fore foot and back foot [124]. Due to the fact that PDWs do not exhibit ankle action/push-off (dorsiflexsion), I hypothesis that this draw-back can be aliviated by shaping the ROS so that it is able to push the walker forward similar to dorsiflexsion.

For decades, PDWs have been constructed and mathematically analyzed, ameliorating and varying their physical designs and analytical models. Because of the diversity and range of past PDW research, I am not able to discuss every single permutations and combinations of past research in this field. Therefore I am presenting all past research of two dimensional passive (un-actuated) dynamic walking I was able to find in Table 2.1.

Note that my summary of past research in Table 2.1 shows much capacity for further exploration in this field. No one single PDW model or PDW study exists that encompasses all, or even most of the areas presented in Table 2.1. Although much has been acomplished in the area of PDWs, noone was able to encompass and progress current designs. A large part of my dissertation will concentrate on filling these unexplored areas and will be discussed in later sections. The highlighted areas in Table 2.1 are able to be filled with my developed PDW model, discussed in a later chapter.

Table 2.1: Two Dimensional Passive Dynamic Walker Research

Plain Antrop. Mass Dissimilar Ground Joint Siff. Human Gait Walker Distribution Sides Kinetics or Damp. Research