Robotics Application of a Method for Analytically Computing Infinitesimal
105
Despite the progress of exoskeletons and biped robots to date, their movements
remain in general less robust and adaptive compared to those of humans. The robustness
and adaptation of human walking is due to the structure and function of the neural control
networks within the nervous system. Decades of study in neuroscience and biology have
begun to uncover how these controllers operate [6]. Controlling our exoskeleton with a
computational model of these networks, which we call a “Synthetic Nervous System”
(SNS) [7], may endow it with more robust and adaptive locomotion.
Our ultimate goal is to control our exoskeleton and FES with an SNS model that
incorporates sensory feedback to coordinate the motion of different limb segments in
a flexible framework that can be altered by descending influences in a task-dependent
way. These two features have been shown to underlie periodic motor output in animals
[8–10]. Neural models of animal locomotion in [11–13] can serve as a basis for the
organization of our SNS model moving forward. However, with the actuators of the
exoskeleton more closely resembling servomotors than antagonistic pairs of muscles,
design of joint movement controls will be more in line with the design process outlined
in [14]. Interjoint coordination will be relying on sensory feedback signals from the
positions of other joints and forces on the leg [11, 15].
An expected hurdle to developing and tuning this SNS will be in the coordination of
the multiple rhythmic systems, i.e. the oscillation of each joint. Inter-joint coordination
arises from sensory pathways between leg joints adjusting the oscillatory phase of the
other joints. For example, loading information from the foot adjusts the phase of the
hip’s control network to generate propulsion [11]. But how should the strength of such
pathways be tuned? One tool that quantifies how periodic trajectories are altered by
perturbations is the infinitesimal phase response curve, or iPRC. An iPRC reveals how
the cycle’s phase changes in response to perturbations applied to each of its state variables
at different phases throughout its limit cycle. By locating the areas of higher and lower
sensitivity, we may be able to design sensory pathways that exploit the oscillators’ phasedependent sensitivity to inputs. In addition, understanding how sensory information
alters the relative phase of the joints, we may be able to determine the stability of periodic
trajectories (i.e. how the exoskeleton walks) and its robustness to system parameters.
Despite the potential benefits these iPRCs could provide, the process of finding the
iPRCs via brute force guess-and-check can require long computation times, especially
as the dimensionality of the SNS model grows. Analytical methods for generating iPRCs
exist, which should be faster and computationally less expensive than brute force methods. However, such methods mostly consider smooth systems and tend to break down
in non-smooth systems due to the Jacobian matrices of the system’s vector fields not
being well defined [16]. Our model joint model system contains piecewise functions and
certain hard boundaries (i.e. “sliding conditions”) that make it non-smooth. However,
more recently developed analytical methods for finding iPRCs can treat systems whose
dynamics contain piecewise functions and sliding conditions [16, 17]. In this work, we
will apply these newer methods with a reduced model of our SNS exoskeleton controller
to compute its iPRC numerically. We compare this iPRC to one generated via brute force
(i.e. repeated perturbed simulation). We then discuss future applications for this work.
105
Despite the progress of exoskeletons and biped robots to date, their movements
remain in general less robust and adaptive compared to those of humans. The robustness
and adaptation of human walking is due to the structure and function of the neural control
networks within the nervous system. Decades of study in neuroscience and biology have
begun to uncover how these controllers operate [6]. Controlling our exoskeleton with a
computational model of these networks, which we call a “Synthetic Nervous System”
(SNS) [7], may endow it with more robust and adaptive locomotion.
Our ultimate goal is to control our exoskeleton and FES with an SNS model that
incorporates sensory feedback to coordinate the motion of different limb segments in
a flexible framework that can be altered by descending influences in a task-dependent
way. These two features have been shown to underlie periodic motor output in animals
[8–10]. Neural models of animal locomotion in [11–13] can serve as a basis for the
organization of our SNS model moving forward. However, with the actuators of the
exoskeleton more closely resembling servomotors than antagonistic pairs of muscles,
design of joint movement controls will be more in line with the design process outlined
in [14]. Interjoint coordination will be relying on sensory feedback signals from the
positions of other joints and forces on the leg [11, 15].
An expected hurdle to developing and tuning this SNS will be in the coordination of
the multiple rhythmic systems, i.e. the oscillation of each joint. Inter-joint coordination
arises from sensory pathways between leg joints adjusting the oscillatory phase of the
other joints. For example, loading information from the foot adjusts the phase of the
hip’s control network to generate propulsion [11]. But how should the strength of such
pathways be tuned? One tool that quantifies how periodic trajectories are altered by
perturbations is the infinitesimal phase response curve, or iPRC. An iPRC reveals how
the cycle’s phase changes in response to perturbations applied to each of its state variables
at different phases throughout its limit cycle. By locating the areas of higher and lower
sensitivity, we may be able to design sensory pathways that exploit the oscillators’ phasedependent sensitivity to inputs. In addition, understanding how sensory information
alters the relative phase of the joints, we may be able to determine the stability of periodic
trajectories (i.e. how the exoskeleton walks) and its robustness to system parameters.
Despite the potential benefits these iPRCs could provide, the process of finding the
iPRCs via brute force guess-and-check can require long computation times, especially
as the dimensionality of the SNS model grows. Analytical methods for generating iPRCs
exist, which should be faster and computationally less expensive than brute force methods. However, such methods mostly consider smooth systems and tend to break down
in non-smooth systems due to the Jacobian matrices of the system’s vector fields not
being well defined [16]. Our model joint model system contains piecewise functions and
certain hard boundaries (i.e. “sliding conditions”) that make it non-smooth. However,
more recently developed analytical methods for finding iPRCs can treat systems whose
dynamics contain piecewise functions and sliding conditions [16, 17]. In this work, we
will apply these newer methods with a reduced model of our SNS exoskeleton controller
to compute its iPRC numerically. We compare this iPRC to one generated via brute force
(i.e. repeated perturbed simulation). We then discuss future applications for this work.