Foot placement in robotic bipedal locomotion

Human walking is remarkably robust, versatile and energy-efficient: humans have the ability to handle large unexpected disturbances, perform a wide variety of gaits and consume little energy. A bipedal walking robot that performs well on all of these aspects has not yet been developed. Some robots are versatile, others are energy-efficient, and none are robust since all robots often lose balance. This lack of performance impedes their applicability in daily life. Also, it indicates that the fundamental principles of walking are not adequately understood. The goal of this thesis is to increase the understanding of the mechanics and control of bipedal locomotion and thereby increase the performance of robotic bipedal locomotion. This increased understanding will also be useful for the development of robotic devices that can help people with a decreased ambulatory ability or that can augment the performance of able-bodied persons. Bipedal locomotion is in essence about the ability to maintain control over the position and velocity of the body's center of mass (CoM). This requires controlling the forces that act on the CoM through the foot. The contact forces between the foot and the ground can be manipulated to some extent through ankle torques or upper body motions, but are mostly determined by the location of the foot relative to the CoM. The limited influence that ankle torques and upper body motions have on the contact forces and consequently on the CoM is best illustrated when one tries to remain balanced on one foot without taking a step. When slightly perturbed, balance is quickly lost and a step must be taken to prevent a fall. This demonstrates that balance control in walking relies on adequate control of foot placement (i.e., the location and timing of a step), which therefore is our main focus in the control of robotic gait. The focus on foot placement control is different from other popular control approaches in robotics. In ZMP-based control, one typically adjusts the robot's state to achieve a predefined foot placement. In Limit Cycle Walking, passive system dynamics mostly determine foot placement. This thesis presents foot placement strategies that can be adapted both in step time and step location, are an explicit function between the initial robot state and the desired future robot state, and are computationally relatively inexpensive to allow for real-time application on the robot. The contributions of this thesis to bipedal walking research are: a theoretical framework, simulation studies, and prototype experiments. These contributions provide insight in how foot placement control can improve the robustness, versatility and energy-efficiency of bipedal gait. Regarding robustness, this thesis introduces the theoretical framework of capturability to analyze or synthesize actions that can prevent a fall. Fall avoidance is analyzed by considering N-step capturability: the system's ability to eventually come to a stop without falling by taking N or fewer steps, given its dynamics and actuation limits. Low-dimensional gait models are used to approximate capturability of complex systems. It is shown how foot placement, ankle torques and upper body motions affect the CoM motion and contribute to N-step capturability. N-step capture regions can be projected on the floor: these define where the system can step to remain capturable. The size of these regions can be used as a robustness metric. Regarding versatility, this thesis derives foot placement strategies that enable the system to evolve from the initial state to a desired future state in a minimal number of steps. Simulations on simple gait models demonstrate how these foot placement strategies can be used to change walking speed or walking direction. Regarding energy-efficiency, we learn that simple gait models demonstrate human-like foot placement strategies in response to a stumble when optimizing for either one of the following cost measures for foot placement: peak torque, power, impulse, and torque divided by time. For robotic control, these results indicate that actuator limitations should be taken into account in the execution and planning of foot placement strategies. Regarding robot experiments, we integrate the concepts from the capturability framework into the control of a robot. The low-dimensional gait models are shown to be useful for the robust control of a complex robot. The model takes only the CoM dynamics with respect to the center of pressure (CoP) into account. The application of this model together with force-based control strategies lead to robust robot behavior: upright postural balance is maintained when the robot is pushed and one of the feet is placed on a moving platform. Successful application is also shown for single legged balancing with compensatory stepping to regain balance after a push and (simulated) walking. The main conclusion is that analyzing walking control as a combination of decoupled and low dimensional control tasks allows us to derive simple and useful control heuristics for the control of a complex bipedal robot. We find that the key control task is foot placement, which mostly determines the system's CoM motion by defining possible CoP locations. We can approximate the set of possible foot placement strategies that will not lead to a fall. This set specifies the bounds to which foot placement strategies can be adjusted to achieve more versatile or energy-efficient behavior.