Variable Inertia Reaction Wheel Biped

Variable Inertia Reaction Wheel as a Balance Assistive Device for the Elderly

Daniel Piedrahita

Reducing falls in the elderly is one of the urgent challenges of aging societies. At least one third of the senior population over the age of 65 falls each year in the United States, leading to over 9,500 deaths and 800,000 hospitalizations yearly. Balance assistive devices have been developed and studied in the past, with promising results in reducing falls. Though these studies have had success in preventing falls, both devices are heavy and cumbersome to wear. As a solution to these issues, we propose a variable inertia reaction wheel as an improvement to existing balance assistive devices. A variable inertia reaction wheel has the advantage of a smaller required package, with the device only increasing in size during falling events when large stabilizing torques are necessary.

Dynamics Model

To quickly and cost-effectively evaluate the feasibility of the variable inertia reaction wheel, a simulation of a human bipedal walking system is developed. Limit cycle walking gaits are generated for this system through trajectory optimization methods. These open-loop optimal trajectories are then feedback stabilized using a time-varying linear quadratic trajectory regulator. This controller is then validated on a full physics simulator and disturbances are applied as horizontal pushes to the walking system’s hip.


manipulator equations

Trajectory Generation

Optimal trajectory generation methods are utilized to generate realistic walking gaits on the bipedal walking system. Direct collocation is used to generate position and velocity trajectories for each of the joints, as well as nominal inputs to generate these joint trajectories during one step of walking. The gait generation problem is posed in the following formulation:

trajectory optimization framework

With the following constraints and objective function:


objective function

Trajectory Stabilization

For generated optimal walking trajectories to be robust to disturbances and model uncertainty, a time-varying linear quadratic regulating feedback controller is designed which regulates joint angles to their nominal joint trajectories. To compute the feedback controller, the system is linearized around the nominal trajectory:

linear system.PNG

Then, the differential ricatti equation is solved backwards in time:

differential ricatti equatin.PNG

In this case, the R matrix is chosen to simulate an elderly person. Feedback torques at the knees and hip are heavily penalized, while the feedback torques of the reaction wheel are not penalized, encouraging the system to use the reaction wheel to reject disturbances. This feedback controller is able to effectively track walking trajectories:


Human Gait Comparison

Visual analysis shows that the walking gait motions qualitatively closely match human motion during walking. In addition, the cost of transport of the simulated system was generally between 0.38-0.42, which closely matches the human optimal cost of transport of 0.39. Because of this, we believe that this formulation was able to effectively capture the important elements of human bipedal locomotion.

Disturbance Rejection Improvement

Once a fully automated process for generating optimal walking gaits and feedback controllers for arbitrary walking speeds and disturbances was developed, a study was performed to evaluate the performance improvement of the variable inertia reaction wheel to human locomotion. A summary of this study can be seen in the following images. In all three images, a 75 Newton push is applied to the hip of the walking system.

No reaction wheel. The system falls when a 75 Newton push is applied to the hip.
Fixed Inertia reaction wheel. The system is able to reject the 75 Newton disturbance with a small stumble.
Variable Inertia reaction wheel balance assistance device. The system is able to reject the 75 Newton push with no problems.


This figure displays the maximum disturbance rejected for each of the three studied systems at various walking speeds. The walking system with no reaction wheel is unable to reject large disturbances, and cannot even walk effectively at higher speeds, which accurately simulates the fragile walking abilities of elderly people. Disturbance rejection capability is greatly increased with the introduction of a reaction wheel balance assistance device, and even further improved by the variable inertia reaction wheel device. This experiment suggests that a reaction wheel is an effective device for preventing falls in the elderly. More interestingly, the inclusion of a variable inertia reaction wheel mechanism provides increased disturbance rejection capabilities over a fixed inertia device.