PhD Student @ Carnegie Mellon

Funneling Swing Leg Motion into Target Positions

INTRODUCTION

Understanding the background to why and how robots do what they do is an important part in robotics research. A strong focus in robotics is bipedal locomotion, walking. There are many ideas behind how to effectively make a robot walk and one of the trains of thought are to use passive elements to do most of the walking while controlling only certain things to make up what still needs to work, with active control.

One of the ways roboticists research walking is by studying humans or animals to see how they walk and to understand the phenomena in general. There are principles to be learned from biological organisms by studying how they move that can potentially be harnessed to create more robustness in walking robots. One rather successful method to make robots walk is to create a trajectory for the robot to follow and then control the robot to follow it. This method and many adaptations to this method are promising. Another form of developing walking behavior is to have the simple passive elements create the majority of the behavior. This passive dynamic walking is possible in bipedal robots when the robot is place on an inclined plane, but is also a promising route for creating the behavior. One exciting aspect of creating passive dynamics is that there is the potential for the passive dynamics to actively adapt to the surrounding environment, meaning that if the robot is pushed in an unexpected way the passive elements have the potential to adapt to the situation.

Along with studying the pure motion of walking, many scientist study specifically how the human body creates the walking behavior to garner insight into effective methods for controlling walking. The progress in this method has led to models that are based on muscles and neurons that actuate and create the walking behavior. This type of understanding therefore lends itself to not only help develop better walking robots, but to also aid in building better prosthesis for humans. Understanding the complexity of motion and how the motion is created from the neurons down to the muscles might lead to robotic prosthesis that are able to act upon the neural signals in accordance with how a natural human body might, given that the models are developed enough to fully interpret the signals. These exciting potential developments make the study of basic human motion a must.

The background of how the full models of walking in humans and robots started with simple models of inverted pendulums. If an inverted pendulum with the pivot point anchored to the ground glides along its natural motion, it is easy to see that this motion is similar to a step that a human makes with the foot on the ground. This model provides similar dynamics to human walking on the ground, meaning that the ground forces are similar between human walking and the model of the inverted pendulum. Using the same ideas of inverted pendulums, more predictive models have been established that predict the foot placement during walking. This is accomplished by keeping the pendulum body trajectory linear instead of the arc it would naturally make and then observing where the foot would need to be placed to keep this trajectory. Since in humans the vertical motion is small compared to the horizontal motion in walking this assumptions is reasonable and provides a good way to predict where a humanoid robot would need to take it’s step. Another observation of human motion has led to adding springs instead of a massless bar for the inverted pendulum since a compliant behavior in the limbs can be observed from walking and running. Adding this type of spring behavior to the inverted pendulum, modeled even better the ground reaction forces meaning that the models were approaching human walking behavior and a real understanding of the walking was being captured. Taking this idea further researchers starting modeling the multiple limbs, called segments in the modeling language, with the same inverted pendulum idea. These segmented legs could then incorporate the same principles of a single segment inverted pendulum, with a spring, by producing the vertical behavior by bring the joint angle of the knee in tighter. Creating the same springiness in the segments during walking was then accomplished by not only using passive elements, but then using the somewhat passive elements that humans use, muscles. While muscles are not completely passive, there are reflexes from forces and stretching that created a passive control in muscles. This passive control of muscles with reflexes was then able to not only predict the ground forces, but was also able to capture the similar muscle activation patterns in humans by providing similar spring-like reflex feedback with muscles.

The ground reaction forces are good indicators of the dynamics of the walking system being captured, this primarily pertains to the stance leg, or the leg on the ground during the period of walking. The swing leg is less understood, but the muscle activation patterns are good indicators of whether the dynamics of the swing leg are being captured. While some reflexes of the swing leg have been captured there are mismatches between the muscle activation patterns between the muscles in the swing phase in humans versus what the model predicts.

In the previous figure from Geyer, 2010 it can be seen that in the swing phase after the dotted line there is a discrepancy in the amplitude of the hamstring activation and also a missing activation of the vasti group muscles.

Due to the discrepancy in the muscle activation patterns this work attempts to work from the basic dynamic pendulum model of the swing leg and simple passive springs to look at the basic behavior of the swing leg to attempt to understand what role the passive elements play and what part needs to come from active control. Also a patterns the dynamic behavior are studied to capture principles of the basic motion that is needed in swing leg motion.

METHODS

There are two parts to the model. The first is the dynamic model of two masses and two massless bars attached to ground with a rotational joint and attached to each other with a rotational joint. The masses are attached at the knee and at the foot. This arrangement for the swing leg is basically a simplified model and just a double pendulum, which is a chaotic system. To analyze this system’s natural motion the potential energy of the system for each point the foot could be in was determined. Because both masses are considered in this system and there are constraints on where the masses can be with respect to each other a relationship between the masses and foot position was needed. This relationship was found by relating the foot position with the joint angles. There is a hip angle alpha which starts at vertical position and is positive in the clockwise position. This is intuitive when considering a human system because in the same way that the hip muscles connect to the trunk of a human which remains mainly upright so the hip joint angle connects to the vertical upright position. The knee angle gamma starts from the thigh and is positive as it rotates co unter-clockwise.

The relationship between the masses positions and the foot positions in joint angles is as follows. The primary joint angles are alpha and gamma. Some helpful angles to define are eta, beta, epsilon, and alpha’ where beta is the angle between radius of the hip joint to the foot and the thigh, eta is the angle between the radius and the vertical and epsilon is the angle between the horizontal and the thigh in the counter-clockwise direction starting from zero degrees. Alpha’ is the angle from the horizontal to the thigh in the clockwise direction. The relationships are below.

Once there are joint angles defined for every foot position this the potential energy of the system can be found as functions of the joint angles.

The potential energy for the dynamic system is in [figure X]. From the potential energy of the mass system it can be seen that the foot naturally doesn’t want to go forward and upward and so already tends to push the foot towards a specific foot placement position. Unfortunately it does sink upwards and to the left hand quadrant which is not typical of human motion in walking, meaning that there are some other elements that are there to keep the foot from going in that direction. The simulation of the model was implemented in the multibody dynamic environment of Matlab, Simmechanics.

There are two blocks for the masses and there are two blocks for the joints. The hip joint is attached to ground. Springs are added by sensing the angles with body sensors and passing them through the spring equations and then passing them to joint actuator that actuate the joints.

Where these elements are passive or are active control are unclear from this point.

The next step is to therefore add other elements to the model that represent the human physiology, muscles. Instead of adding the complex behavior of muscles initially, the muscles are added as springs. Since not only is some of the behavior naturally similar to spring like behavior, but can be controlled with force feedback to even more represent springs. The springs added are rotational springs about the hip and knee joint. The parameters of the springs that are added and vary the behavior of the swing leg are the spring constants, the rest length of springs and the moment arms of the rotational springs at each joint. Changing the spring constants changes how strong the strength of muscles and their contribution to the system. The rest lengths are the angles at which the springs engage instead of just being slack. These rest lengths are referred to as reference angles as these are rotational springs being considered. The reference angles change the potential field by shifting the energy to specific positions. The moment arms of the springs are just gains for the uni-articular muscles, the muscles that span only one joint, but in the bi-articular muscles, the muscles that span two joints, there can be a different contribution of the muscle for to each joint depending on the moment arm which can change the behavior. The same type of muscle motion could create a large force at the hip with a larger moment are at the hip, but create less of a force at the knee with a smaller moment arm for a biarticular muscle.

The potential energy of the springs and how they change the overall potential field of the system together then can then be explored.

The combination of the muscles and the springs which specific parameters can create a passive motion that ends at a specific point in the same way that the foot would start from a specific position push off and wind up at specific location.

When considering the linear inverted pendulum model that predicts the foot placement to create a stable walk performance of bipeds a target position would be desirable in the combined potential field. This would reduce the excess energy required to get the foot to the position it needed if the springs and the gravitational potential were working in part to passively place the foot in the right position. With some of the work being done by passive elements then less would need to be controlled by the body actively.

In order to determine the configuration of the muscle parameters that would best create natural swing leg behavior the muscles were first hand tuned and then were tuned by providing an example potential function and attempting to optimize the muscle parameters to the created potential function.

The cost function was therefore the difference between the example potential field and the muscle potential functions, squared. In order to create a example potential field there are some ideal characteristics of the muscle potentials were defined. First the potential field was set up so that it could start from different take off positions, but would still stay within a given area and then head down towards a specific position. The behavior was therefore a sort of funnel with high side walls that the foot would not be able to overcome and a slant downwards toward where a foot would need to be to end the swing. This was created by defining several Gaussian ellipses rotated and shifted in the field to create the high side walls. Then there was a intentional slope added to the inside of the funnel. The Gaussian ellipses were chosen because the decay of the walls could be controlled and a smooth transition between the additive fields could be found. The general equations of the Gaussian ellipses are:

RESULTS AND DISCUSSION

The first trial with optimizing the cost function of the example potential function and the muscle potentials was on a direct funnel or an attractor field going directly down to a point.

Unfortunately, the potential field did not optimize to an acceptable one.

With the simple attractor field optimization not working well, the previously discussed field was used for optimization.

One again the potential field did not optimize to what would have been an acceptable field.

Several problems arose during the optimization step that still need work. The first is that the signs of the muscles spring constants change which means that the muscle provides the wrong torques which is physiologically impossible when considering human muscles and human walking. Another problem with the optimization is that the it set some of the muscles’ reference angles to random angles which the muscles cannot physiologically reach and some potential fields are not even in the correct range of motion with a leg going completely vertical upwards. This is equivalently making the muscle inactive completely in the acceptable range of motion. Another complication is that there was no muscles defined for certain potential field position that would be able to account for unwanted motion. This means that other muscles in the legs need to be considered to get better results for human motion as we know that stable walking and running is possible with certain muscles engaged.

Therefore the optimization was not the best way to start the analysis and an exploration or parameters with hand tuning was undertaken. Without any tuning the natural motion of the double pendulum is as follows.

The first muscle that was tuned was the hamstring which showed a sharpening on the foot position to a point and then brings the foot back to a starting position. Adding the hip flexor muscle balances out the effect of the hamstring and keeps the energy from going to high on the opposing side of the field. The hamstring have exactly opposing characteristics and the combination creates a relatively stable position alone.

CONCLUSION

The dynamic system of double pendulums with springs was analyzed for the passive properties that would reach a target foot position. An optimization of the parameters was undertaken, but was put on hold until there is more basic understanding of the dynamic system.

The future work is determine the role of each of the muscles in the system, to find appropriate bounds for the acceptable range of muscle behavior, spring constants and reference angles, for the cost function of the optimization. More work is needed to identify the muscles that are active during different perturbations of the system.