Control of 3D dynamic walking in bipedal legged machines or humanoid robots remains a challenging problem. To address the complexity brought by the high degrees of freedom (DoFs) in the target system, very often a simple template model is used as an intermediate bridge. A template model is typically much simpler, but can still capture the key characteristics of the dynamics of a particular motion. Usually, a control policy is first developed to regulate the behavior of the template model, then resultant trajectories of the template model are mapped to the whole-body motion of the more complex robot through some optimization-based task-space control strategies. Currently, the most widely used walking template is the Linear Inverted Pendulum Model (LIPM). It has gained popularity due to its simple linearized dynamics, and walking controllers based on the LIPM have had some success. However, the resultant walking gaits from the LIPM miss some of the dynamics that can lead to more efficient gaits such as those in a human. This is due in part to its constant height assumption and the lack of a double support phase.
Recently, an alternative walking template called the Dual-SLIP (spring-loaded inverted pendulum) model (or Bipedal SLIP model) has been proposed. This model has its roots in biomechanics studies. With leg compliance it can faithfully reproduce the ground reaction force (GRF) patterns and center of mass (CoM) vertical oscillations observed during human walking. Also its bipedal nature allows it to seamlessly integrate the double support phase during walking. It even has the potential to become a general template that can model both walking and running. Some analyses and control strategies have been developed around this new template. However, almost all the studies are limited to the 2D version of this model.
The behavior of the 3D Dual-SLIP model is much more difficult to regulate than its 2D counterpart. And the problem is accentuated when walking over uneven terrain, especially with step height uncertainty. However, to actually use this model as a walking template for a humanoid, the 3D extension is necessary. With such motivation, this thesis is dedicated to develop a dynamic walking controller for a humanoid based on the 3D Dual-SLIP model while traversing uneven terrain.
The development is divided into three stages. In the first stage the focus is on 3D walking over flat ground. A new optimization-based approach is proposed to find periodic walking gaits. Through analysis of the dynamics of the 3D Dual-SLIP model, a novel symmetry condition has been identified that is used to greatly simplify the optimization process. It has also been found during the study that the regular conservative 3D Dual-SLIP model is not capable of generating forward walking at speeds faster than 1.4 m/s (assuming a leg length of 1 m). To address this limitation, a variant of the regular Dual-SLIP model, the actuated Dual-SLIP, is introduced to address high-speed walking. With bio-inspired leg actuation, the model can walk up to 2 m/s, which is at the top end of the range of human walking speeds. Since the periodic gaits of the 3D Dual-SLIP model are found to not be self-stable, a discrete-time infinite-horizon LQR controller has been developed to regulate the state of the model. Through this approach, the controller can recover from significant disturbances by automatically adapting footstep positions during walking.
In the second stage, the development of the first stage is extended to known uneven terrain, which is the first time the 3D Dual-SLIP model (with leg actuation) has been used to generate dynamic walking gaits over uneven terrain. Since the symmetry condition can no longer be used to aid in the search for periodic gaits over uneven terrain, an improved CoM trajectory and footstep position generation method is developed based on multiple-shooting optimization that is applicable to both flat ground and uneven terrain (with elevation changes up to +/-10% of leg length per step). The resultant gait for flat ground is consistent with that found with the simpler approach of the first stage. The gait over uneven terrain also shows a rich set of human-like characteristics as observed in biomechanics studies.
Finally, the work in the second stage is further extended to address a more challenging problem: 3D "blind'' walking with no knowledge of terrain information. Through the expansion of the techniques introduced in the second stage, swing leg retraction and extension strategies (towards the end of the swing phase) for the 3D actuated Dual-SLIP model are developed that allow it to automatically adapt its walking gait over unforeseen terrain height changes up to +/-5% of leg length while maintaining full forward speed.
The resultant CoM trajectories and footstep positions of the 3D Dual-SLIP model from the development in all three stages are used to successfully orchestrate dynamic walking motion in real-world humanoid robot models in simulation through a task-space control framework. This is the first demonstration of Dual-SLIP based dynamic walking in a humanoid.