# Stable Heteroclinic Channel-Based Movement Primitives: Tuning Trajectories Using Saddle Parameters
## 1 Introduction  
To develop new versatile, robust, learnable control frameworks for robots, we are seeking ways in which a heteroclinic system can be adjusted _after_ learning _while_ remaining stable. Most controllers are based around \"PID\" parameters (Proportional Integrative and Derivative gains) that describe linearized stability around a single stable point (the goal state). Heteroclinic systems orbit not just one, but multiple equilibria, which is useful in controlling periodic motions such as robot locomotion gaits, repetitive motions in manufacturing, or more complex trajectories. In complex, high-dimensional spaces, equilbria can represent holding behaviors, system status patterns, or action sequences. By visualizing these behaviors in Cartesian space [1], we hope to better understand the underlying dynamics. The robustness of such systems comes from their proven stability [2; 3; 4] and the ease of use can come from learnability [5; 6; 7]. The ability to combine multiple states, which can be thought of as primitive building blocks, enables an inherent versatility to combine and recombine different states. Here, our goal is to increase the versatility further by demonstrating that state-to-state trajectories can be tuned by parameters associated with each state. Creating behaviors as a composite of stable primitive building blocks has been a powerful tool for controlling robots. For example, Dynamic Movement Primitives (DMPs) have been effective for over twenty years in providing learnable, smooth kinematic control policies [8; 9; 10; 11; 12]. DMPs have modular components called kernels, which are generated from stable points that are activated in time sequences. If those stable points are replaced with saddle points, such that the unstable eigenvector of one saddle points to the next saddle, we can refer to them as stable heteroclinic channels (SHCs) [7; 13; 14]. Using SHCs to generate the kernels results in comparable performance to DMPs, and has the added benefit of visualization [1] and the potential to enable the timing of the saddle-to-saddle transition to be embedded in the dynamic system. Saddle equilibria-based dynamical systems can help engineers better bridge gaps between neurobiology and biologically inspired artificial intelligence. The spectrum of biologically inspired controllers ranges from high-level, behavior-based controllers like finite state machines, where a robot's desired action is encoded into the control software [15; 16], to low-level neuromorphic controllers like neural networks, where neuronal functions are interconnected at scale [17; 18; 19]. SHCs are intermediary. Because the connectivity is in the mathematically constructed connection matrix, they can be analyzed. Yet, they are abstract enough that the system dynamics are less prone to the \"black-box\", unexplainable dynamics seen in high-dimension frameworks [20; 21]. For this reason, biologists have used SHCs to model the population behavior of biological neurons because they abstract the dynamics of the system into tractable components [22; 23] and engineers are starting to use SHC-based movement primitives (SMPs) to control bio-inspired robots [1; 14; 24]. Specifically, such bio-inspired controllers make sense in high-dimensional systems where predictive forward models are unavailable. Examples include compliant robots [25; 26] in complex, dynamic environments [27; 28; 29]. The current alternatives for systems like these are model-based controllers and high-dimensional \"black-box\" controllers. Model-based controllers are computationally expensive [26; 30; 31] and thus not optimal for mobile robots with limited on-board computation, and \"black-box\" controllers lack explainability when the robot's behavior is scrutinized [32; 33]. This article builds on the SMP parametric transparency that was introduced in [1], especially for the saddle value, \(\
u\), within several orders of magnitude.