Spring-Loaded Inverted Pendulum model

In general, good engineering results in designs that are as simple as possible, yet the AMASC is a reasonably complex mechanism. The motivation for the behavior of the actuator (series springs with adjustable stiffness) is gained from observations of animals, and the complex design is the simplest we could create that is capable of implementing the desired behavior.

Behavior

Animal running is comprised of the knees, hips, and ankles all working together to create a mostly linear spring with a joint at the hip, much like a pogo stick with a mass at the top. This is represented by the Spring Loaded Inverted Pendulum (SLIP) model, which has been shown to accurately model the motion of the center of mass of running animals. The control inputs to this simplified running model are hip angle at touchdown, spring rest length, and spring stiffness. Animals use all three control inputs in various situations, both to regulate a gait over verying terrain and to change their gait on consistent terrain. For example, animals adjust leg stiffness to accomodate surfaces of varying stiffness, such as a transition from grass to pavement. When running over consistent terrain, animals can use either stride length or stride frequency to control forward speed - and stride frequency is controlled through leg stiffness adjustment. Most robots do not have the capability of controlling leg stiffness, but also are not nearly as capable or as energetically efficient as animals.

Implementation

The AMASC in motion

Implementing spring-like behavior with controllable stiffness is not an easy task. The most obvious approach to most people is to use a computer-controlled gearmotor, because the software is very flexible, and can easily be changed or adjusted so the motor exhibits the desired behavior. The problem with this approach is that electric gearmotors may not do what the software tells them to; they have high rotational inertia, which limits acceleration, and insufficient power output for running and jumping on a mobile robot.

Air springs are difficult to control and inefficient, hydraulics are very heavy and power-intensive. Both approaches generally require offboard compressors and hoses leading to the robot, which is highly undesireable in a mobile robot. Exotic actuation methods such as artificial muscles have not yet become feasible for robotic applications.

The fiberglass springs

The AMASC design is a mechanical implementation of the SLIP model. Rather than simulate a spring through software, the AMASC has a physical spring. An electric motor controls the rest length of the spring, which moves the knee joint when no external loads are applied. A much smaller motor controls the joint stiffness. We have effectively designed the ``natural dynamics'' of the system such that the behavior, even with the motors turned off, is that of the SLIP model. With a hip, the AMASC would have mechanical control of the same three control parameters of the SLIP model: leg stiffness, spring rest length, and hip angle.

The spiral pulleys modify the spring function

The primary goal of the AMASC was to match the SLIP model as closely as possible, so the mechanism can easily be represented mathematically as part of an integrated software/mechanical control system. Related design goals include zero backlash and very low friction. In addition, low mass is important for a mobile robot. The spring function of the knee is created by a set of shaped pulleys.

Control System Design

The control system is designed around the mechanical model, and is intended to do two basic things. The first is to adjust the mechanism configuration so its physical properties match the commanded spring stiffness and rest length. The second is to actively control the set point motor to simulate the proper settings when the mechanism is out of adjustment. In the ideal situation, the motor will have to do very little work, and will allow the mechanical springs to store and release most of the energy in a running gait.

Adjusting the spring stiffness is accomplished with a simple PID position controller on the pretension motor. Since the position of the pretension motor corresponds directly to a stiffness felt at the leg, no further control is required.

Adjusting spring rest length is accomplished using a simple PD controller on the position of the set point motor, with some added complexity when the pretension is not properly adjusted. In the ideal case, the motor position is fixed at the desired set point, and the spring physically matches the desired spring stiffness. To simulate this system, we must control the motor position so that it will simulate the desired spring stiffness if the physical system does not match our desired system. The torque on the leg applied by the physical system should match the torque on the leg applied by the desired system, by setting the equations for both torques to be equal to each other, we can solve for the desired motor position. We then apply a PD controller to the set point motor to move it to the desired position, along with a spring cancellation force to hold it against the force applied by the springs. With the spring cancellation force, the PD control can adjust the motor position as if it were an independent mass, without the attached spring and associated dynamics.

Performance

A mathematical model of the actuator system

Measuring performance of a system requires some metric, such as top speed, or efficiency, or a comparison to some goal. In our case, the performance metric is similarity of the hardware prototype to the mathematical model. Our model is relatively simple, and can be incorporated as part of a complex control system. If the mechanism behavior matches that of the mathematical model, then our robotic system has a much higher likelihood of being controllable.

Comparison of simulated AMASC and prototype AMASC

We tested the actuator by commanding the spring set-point to move suddenly, and then measured the response of the knee joint. As expected, the knee joint bounces back and forth a few times, slowly settling to a new position as friction removes energy. The behavior of the model and that of the physical prototype are esentially identical.