Chapter 1

Introduction

1.1 Objectives of Legged Locomotion and Challenges in Controlling Dynamic Walking and Running

The most effective type of locomotion in rough terrains is legged locomotion. During the past three decades enormous advances have occurred in robot control and motion planning of dynamic walking and running locomotion. In particular, hundreds of walking mechanisms have been built in research laboratories and companies throughout the world. The desire to study legged locomotion has been motivated by the need to assist people with disabilities to walk and replace humans in hazardous environments. Underactuation, impulsive nature of the impact between the lower limbs and the environment, the existence of foot structure and the large number of degrees of freedom (DOF) are basic problems in controlling legged robots. Underactuation is naturally associated with dexterity. For example, headstands are considered dexterous [1]. In this case, the contact point between the body and the ground is acting as a pivot without actuation. The nature of the impact between the lower limbs of legged robots and the environment causes the dynamics of the system to be hybrid and impulsive. The impact between the foot and the ground is one of the main difficulties in designing control laws for walking and running robots. Unlike robotic manipulators, legged robots are always free to detach from the walking surface, thereby leading to various types of motions. Finally, the existence of many degrees of freedom in the mechanism of legged robots causes the coordination of the links to be difficult. As a result of these latter issues, the design of practical controllers for legged robots remains to be a challenging problem. Also, these features complicate the application of traditional stability margins. Consequently, the major issues in the control of dynamic walking and running are as follows:

1. Limb coordination. Legged robots are high degree of freedom mechanisms, and consequently, coordination of their links to achieve dynamic walking and running locomotion is complex. 2. Hybrid nature of locomotion due to presence of impact and liftoff. The presence of impact, foot touchdown, and liftoff leads to models with impulse effects and hybrid systems consisting of multiple continuous and discrete phases. In particular, mathematical models describing the evolution of legged robots during walking and running include both discrete and continuous phenomena. Instantaneous discrete phases arise when feet impact the ground or feet liftoff the ground, whereas ordinary differential equations based on classical Lagrangian mechanics describe the evolution of legged robots during continuous phases of locomotion. 3. Underactuation. During certain phases of walking and running such as single support (one leg on the ground) in walking and flight (no leg on the ground) in running, legged robots have fewer actuators than degrees of freedom. 4. Overactuation. During double support phase of bipedal walking (both legs on the ground), biped robots have fewer degrees of freedom than actuators. Due to overactuation, the control input corresponding to a specific trajectory in the state space is not unique. 5. Inability to apply the Zero Moment Point criterion. Most past work in the literature of legged robots emphasizes the quasi-static stability criteria and flat-footed walking based on the Zero Moment Point (ZMP) [2–16] and the Foot Rotation Indicator (FRI) point [17]. The ZMP is defined as the point on the ground where the net moment generated from ground reaction forces has zero moment about two axes that lie in the plane of ground [3]. The ZMP is contained in the robot's support polygon, where the support polygon is defined as the convex hull formed by all contact points with the ground. The ZMP criterion states that when the ZMP is contained within the interior of the support polygon, the robot is stable so that it will not topple. Thus, in this kind of stability, as long as the ZMP lies strictly inside the support polygon of the foot the trajectories are feasible. If the ZMP lies on the edge of the support polygon, then the trajectories may not be feasible. The center of pressure (COP) is a standard notion in mechanics which was renamed as the ZMP by Vukobratovic [3]. The FRI point is a concept defined when the foot is in rotation with respect to the ground [17]. The FRI is the point on the ground where the net ground reaction force would have to act to keep the foot stationary. Thus, if FRI is within the convex hull of the stance foot, the robot can walk and it does not roll over its extremities, such as the heel or the toe. This type of walking is called as fully actuated walking. If the FRI is not in the projection of the foot on the ground, the stance foot rotates about the extremities. Such an event is also known as underactuated walking. As long as the foot does not rotate about its extremities, the ZMP, COP, and FRI points are equivalent [15] (see Fig. 1.1). In the literature of legged locomotion, a statically stable gait is a periodic locomotion in which the robot's center of mass (COM) does not leave the support polygon. A quasi-statically stable gait is a periodic locomotion in which the COP of the robot is within the interior of the support polygon. Moreover, a dynamically stable gait is a periodic locomotion where the robot's COP is on the boundary of the support polygon for at least part of the walking cycle [18]. Thus, during dynamic walking and running cycles, the location of the robot's COP is on the boundary of the support polygon and, as a result, this will prohibit the use of the ZMP criterion. To make this notion more precise, the ZMP criterion is a sufficient and necessary condition for the stance foot not to rotate. However, this does not imply that the walking motion is asymptotically stable in the sense of Lyapunov [18, Chapter 11]. 6. Lack of algorithms to achieve feasible period-one orbits and limit cycles. The main problem in control of legged locomotion is how to design a feedback law that guarantees the existence of a stable limit cycle for the closed-loop system. Underactuation and unilateral constraints must be included in order to design a feasible periodic orbit for legged locomotion. Unilateral constraints are constraints on the state and control inputs of the mechanical system that represent feasible contact conditions between the leg ends and the ground. In particular, leg ends, whether they are terminated with feet or points, are not attached to the ground. Hence, the ground reaction forces must lie in the friction cone to prevent slippage and foot liftoff. Thus, normal forces at the leg ends can only act in one direction, and are unilateral. In addition, if the foot is to remain flat on the ground and not rotate about its extremities, then the FRI must be between the heel and toe, a condition that can be expressed as a pair of unilateral constraints. These facts combined with underactuation during the single support and flight phases complicate the design of motion planning algorithms to generate feasible periodic locomotion. 7. Conservation of angular momentum about the robot's COM during flight phases. During flight phases of running, conservation of angular momentum about the robot's COM is a nonintegrable Pfaffian constraint which complicates the path planning and control of the robot's configuration during landing (flight to stance phase).

Figure 1.1 Two planar bipedal models to compare the COP, ZMP, and FRI points. The FRI point is a point on the ground contact surface, within or outside the convex hull of the foot support area, at which the resultant moment of the force/torque impressed on the foot is normal to the surface [17]. In the left figure, the foot does not rotate about its extremities, thus, the ZMP, COP, and FRI points are equivalent. At the right figure, the foot is starting to rotate since the FRI point is outside the convex hull of the stance foot. We note that the COP is at the tip of the stance foot about which the foot rotates.

1.2 Literature Overview

1.2.1 Tracking of Time Trajectories

Most existing control algorithms in the literature of legged robots are time-dependent approaches based on tracking of predetermined time trajectories generated by the ZMP criterion, inverted pendulum model and nonlinear oscillators as central pattern generators of the spinal cord. The ZMP criterion [2–16] has been used for trajectory tracking in ASIMO [2] and WABIAN [5, 6]. The Linear Inverted Pendulum Model (LIPM) [19, 20] and ZMP criterion-based approaches for stable walking reference generation have been reported in the literature. In these techniques, generally, the ZMP reference during a stepping motion is kept fixed in the middle of the supporting foot. Erbatur and Kurt [9] proposed a reference generation algorithm based on the LIPM and moving support foot ZMP. In addition, they made use of a simple inverse kinematics-based joint space controller to test the reference trajectory for simulation of a 3D, 12-DOF biped robot model. By allowing a variation of ZMP over the...